An Improved System of Mnemonics/Multiplication

by Thomas Coglan

Layout 2

MULTIPLICATION.

To arrange a plan for committing the multiplication table to memory, by any other mode than the usual one, may, by some people, be considered unnecessary, as being already sufficiently easy to acquire, without having recourse to mnemonical aids, but others view it as a most formidable task: many children, and even adults of excellent parts, find it extremely difficult to impress on their minds; it cannot be effected but by frequent repetitions, which generally consumes a considerable portion of time, that might be profitably employed in other studies. This, at an advanced period of iife, by forgetting our juvenile exertions, we may not be so sensible of, as the table by habit has become so familiar to us, that we deem its early acquisition as being unattended with trouble, yet, in most cases, it is a work of some labour and pains—the author has had many applications inade to him for a plan to assist the memory in it; the following plan, he thinks, will he found easy and effectual. They, whose memories are sufficiently tenacions without such aids, will, perhaps, reject it. It is only offered as an assistance to those, who think they stand in need of some helps, different from the common method.

It will be seen, by reference to the table, that this plan consists in making mnemonical words of the several products; which words are made into pictures, to be placed on the walls of an apartment, arranged as they are on the diagrams; the pictures may be cut out, or larger ones drawn from them, and actually pinned to the walls, or by gazing on them, and transferring their images to their respective localities, become by that means fixed;—by putting them upon the walls, children can very easily recollect them all, even before they leave the nursery; and afterwards by degrees to teach them the letters that represent the figures, the whole table will become familiar.

To place but one row of pictures on the wall at a time, and cause the learners to repeat them a few times before the second is put up, will be the best mode; and in the same manner, to act with the remainder; and to let them be well acquainted with one wall, before they attempt the second, &c.

They must also distinctly mark the order of their figures, that go across each wall, and down the sides; indeed that ought to be done before the pictures are placed: If the figures were actually put up, they would be found useful.

		First Wall
	2	3	4	5	6
2	4	6	8	10	12
	Roe	Eve	Bee	Oats	Queen
3	6	9	12	15	18
	Ivy	Ape	Tony	Quail	Tub
4	8	12	16	20	24
	Boy	Tin	Tuova	Nose	Nero
5	10	15	20	25	30
	Atys	Ayla	Eneas	Nail	Goose
6	12	18	24	30	36
	Tun	Tule	Hare	Egeus	Goad

 

		Second Wall
	7	8	9	10	11	12
2	14	16	18	20	22	24
	Tray	Toad	Toby	House	Hen	Hair
3	21	24	27	30	33	36
	Ant	Hero	Ink	Mouse	Egg	Maid
4	28	32	36	40	44	48
	Hebe	Moon	Medea	Rose	Aurora	Arab
5	35	40	45	50	55	60
	Eagle	Iris	Earl	Eolus	Lily	Daisy
6	42	48	54	60	66	72
	Um	Ruby	Jar	Ideus	Dove	Can

 

		Third Wall
	2	3	4	5	6
7	14	21	28	35	42
	Equery	Hat	Howe	Mule	Iron
8	16	21	32	40	48
	Tidy	Hare	Gun	Oars	Robe
9	18	27	30	45	54
	Tib	Inca	Guide	Reel	Lyre
10	20	30	40	50	60
	House	Geese	Fars	Ajax	Dose
11	22	33	44	55	66
	Nun	Gig	Zara	Jail	Dido
12	24	36	48	60	79
	Hare	Mead	Rib	Vase	Cain

 

By repeating them a short time in this way, they will have all the advantages of the common mode, for 8 times 5 they will have to repeat is 64, which a little practice will render so familiar, that they will have no occasion to refer to the wall; with this circumstance in their favour, that if they should be uncertain whether they are correct or not, they can at once determine it, by referring to their symbol. It will be necessary to separate the pictures; not to put them upon the walls as they are now united in the diagrams; but to divide the first wall by the eye into five stripes, and place the symbols at an equal distance from each other, for if they were put close together; it would not be easy for the mind to separate them. The same arrangement to be made on the other three walls;

 

		Fourth Wall
	7	8	9	10	11	12
7	49	56	63	70	77	84
	Rope	Lady	Dog	Keys	Cake	Bear
8	56	64	72	80	88	96
	Jove	Deer	Cane	Box	Bow	Pad
9	63	72	81	90	99	108
	Adam	Canoe	Bat	Fox	Fop	Sow
10	70	80	90	100	110	120
	Oaks	Wax	Posey	Ox	Quoits	Shoes
11	77	88	99	110	121	132
	Cocoa	Babe	Pope	Tatoes	Tent	Sign
12	84	96	108	120	132	144
	Beer	Fido	Sow	Suns	Seaman	Usurer

1st Wall

	02	03	04	05	06
02
03
04
05
06

 

2nd Wall

	07	08	09	10	11	12
02
03
04
05
06

 

3rd Wall

	02	03	04	05	06
07
08
09
10
11
12

 

4th Wall

	07	08	09	10	11	12
07
08
09
10
11
12

only that the third and fourth walls hare each six stripes, which difference from the two first, it will be easily seen could not be avoided.

An objection may be made to this plan, by stating the supposed difficulty of making children acquainted with the letters that represent figures; but this will be found on trial perfectly easy, and will thus render them early acquainted with their use, which if they follow the system of mnemonics in its extended sense, will be so necessary to be known by them; if the proper means be employed, a child of tolerable capacity, could commit the whole table to memory in four lessons of half an hour each.

Although for this table, no system can supersede the necessity of trusting to memory for the recollection of it, yet some assistance may be occasionally derived by learners, in parts of the table, by observing a few partial rules.

When 5 for instance is the multiplier, a child could be taught, that when the multiplicand is an even number, such as 2, 4, 6, 8, &c.—that to take the half of such number, and join a cipher to it, gives the product—thus 5 times 6, is easily ascertained to be 30—for the half of 6 is 3, join to it a cipher, it is the sum 3,0—5 times 8? the half of 8 is 4—join a cipher, is 40—When the multiplicand is an odd number, instead of joining a cipher to the nearest half of such number, join a five to it—thus 5 times 7—the half of 7 is 3, and one over—that one is a 5—joining it to the 3, is 35.—

How much is 5 times 9?—the nearest number to the half of 9 is 4, join to it a 5—is 45.—When 5 is named as the multiplicand, it can be always shifted as the multiplier; for 7 times 5 is the same as 5 times 7.—It is scarcely necessary to point out, that when 10 is the multiplier, that a cipher added to the multiplicand, gives the product, 10 times 6—by joining an 0 to 6 is 60, &c. &c.

11 times any number is very simple, by considering it always as a multiplicand; then whatever the multiplier may be (under 10) to put it down twice:

9 times 11, by putting down 9 twice, is	99.
7 times 11, 7 twice put down, is	77.

When 12 is the multiplier, or multiplicand; the usual manner will be found sufficiently easy, as 6 times 12 is found to be 72; by multiplying the 2 of 12 by 6, it makes 12; putting down 2 and carrying 1, and then multiplying the 1 of 12 by 6, and adding the 1 that was carried, to it; it makes 7; which put down by the 2, is 72—or perhaps the following mode, may be more simple in the same sum,—by observing what number the multiplier is above 5—such excess to be the multiplier of the 2 of 12, and then adding the first multiplier to the 1 of 12, gives the sum.

Thus, in the above sum 6 times 12—the 6 is 1 above 5, once 2 (the 2 of 12) is 2, then adding 6 to the 1 of 12 is 7, joined to the 2 already had, is 72.

8 times 12?—8 is 3 above 5—3 times 2 are 6, the 8 added to the 1 of 12 is 9, joined to 6 is 96.

For 10 times 12 the rule has been already given, by joining an O to 12, is 120—but to do 11 times 12, first multiply the 2 of 12 by the 1 of 11, it makes 2, then add 11 to the 2 of 12, makes 13, prefixed to the first 2 makes 132.

12

11

132

12 times 12 in like manner—Twice 2 are 4; 12 and 2 are 14, joined to 4 is 144.

12

12

144

12 times 14?—Twice 4 are 8; 12 and 4 are 16; joined to 8, is 168.

When 9 is the multiplier, it can be always made the multiplicand, then whatever the multiplier is, reduce it a figure, which note in the mind, belongs to the tens place; then subtract it as originally given, from 10; such remainder will be the unit figure, which joined to the figure in the tens place, gives the product.

Thus 8 times 9—take 1 from 8, leaves 7, then take 8 from 10, and 2 remains, join it to the 7 is 72.

7 times 9?—Make 7 one less, is 6—take 7 from 10, and 3 remains; joined to the 6 is 63—here it is obvious that the 9 need not be used in these operations, but merely understood.

9 times 9?—reduce the multiplier 9 one figure, makes 8; take 9 from 10 and 1 remains; joined to the 8 is 81. This plan applies to all figures under 9 (9 inclusive).

These rules for the figures 5, 9, 10, 11, 12, may be of some little service: for the figures under 5—2, 3, 4, there need no rule, as they can be so easily impressed by the common mode, or the mnemonical one.

For the other figures in the table—6, 7, and 8, an exercise of memory will be likewise the best. Or for multiplying them; the plan introduced into the Lancasterian schools may be found useful; by the fingers of each hand being used to effect that purpose:—thus, if asked how much is 8 times 7?—it is resolved (or any sums between 6 and 9) by always considering the number of figures both in the multiplier and multiplicand that are above 5; and then to press down the proper number of fingers on the palms of the hands to represent those figures, which in numbers are the tens belonging to the sum; the other fingers on each hand that are not pressed down, are to be multiplied by each other, which sum belongs to the units, thus the above multiplier 8, is 3 above 5, therefore 3 fingers must be pressed down on the right hand:—the multiplicand 7, is 2 above 5; press down 2 fingers on the palm of the left hand; these 2 fingers added to the 3 fingers of the right hand make 5, equivalent to 50; then, as there are 2 fingers up, on the right-hand, and 3 fingers, up, on the left—they must be multiplied by each other, 3 times 2 are 6, which added to the 50, is 56, the correct sum.

This plan, or something resembling it, is pretty generally introduced into those seminaries; the same calculations may be effected, though perhaps not so quickly by proceeding in the following manner—subtract the multiplier and the multiplicand each from 10; let the remainder of one be multiplied by the other; their product will be the figure belonging to the units place, then subtract from the multiplicand, the remainder that was had from the multiplier;—this second remainder belongs to the tens place, and being joined with the units figure is the correct, sum. Observe that the greater number must be always made the multiplicand, if not, then its remainder from 10 must be subtracted from the multiplier.

Example.—8 multiplied by 7:

Example.—8 From 10 subtract 8 =	2
Example.—8 From 10 subtract 7 =	3
	56

Multiplying 2 by 3 = 6 the units figure.

Subtracting 3 from 8 = 5 the tens figure—joined = 56.

Example II.—8 multiplied by 8:
Example II.—8 10 — 8 =	2
Example II.—8 10 — 8 =	2
	64

10 minus 8, equals 2, the remainder from the multiplicand,
10 minus 8, equals 2 ditto multiplier.
Multiply one by the other, equals 4, the unit figure.
Subtract the lower 2 from the upper 8 leaves 6, the tens figure.

Example III.—9 times 6:
Exam 10 — 9 =	1
Exam 10 — 6 =	4	- 1 × 4 = 4 -9 — 4 = 5 join 4 to the 5 = 54
	54

To perform the operations in the mind of multiplying figures beyond 12, without having recourse to the usual mode of working them on paper or slate, may be, in some cases, desirable; a few examples are given, in hopes that some general rule may be deduced, to render them still more simple.

—In figures between 12 and 20, the multiplier must be added to the right-hand or unit figure of the multiplicand, to which result join a cipher, then multiply the unit of the multiplicand, by the unit figure of the multiplier, and add such product to the sum gained by the first operation; as in this example.

18 multiplied by 15—

15		Add 18 to 5 makes 23, to which join a cipher, =	230
18		Multiply the 5 of 15, by the 8 of 18, equals	40
270		Which added to 230 is	270

Example II.—16 multiplied by 14:

16		14 and 6 are 20, join an 0, is	200
14		4 times 6 are	24
224		Added are	224

Example III.—19 multiplied by 17:

19		17 + 9 = 26, Join an 0 = 260
17		7 × 9 = 63 + 260 = 323
323

Another mode of multiplying the same or similar figures, is given in these two following examples, which method is less useful than the former one, as it only extends to figures under 20: Example IV.

18		Multiply the unit 8 by the unit 5 makes 40, put down an 0, and carry 4, which added to the units 5 and 8, make 17; put down 7, and carry 1 to the 1 of 16 is 2, in all 270.
15
270

17		7 x 6 = 12, put down 2 and carry 4.
16		4 + 6 + 7 = 17 put down 7, and carry 1.
272		1 + 1 = 2, joined to 7 and 2 = 272

---

---

To multiply figures that are between 20 and 100, a little modification, or rather a fuller explanation of the first rule is requisite.

As in that method; so must the multiplier of any sum above 20, be added to the unit figure of the multiplicand; but then the result must be multiplied by the figure which is in the tens place, or left-hand figure of the multiplicand; afterwards proceed as in the first examples; as 28 times 22 will evince.

Example V.

28		22 and 8 are 30, which multiplied by the 2 of 22 makes 60, join an 0 is 600; next multiply the two units, 8 times 2 are 16, added to 600 is 616.
22
696

Example VI.—Multiply 29 by 24:

29		24 + 9 = 33
24		33 × 2 = 66 join a cypher = 660
696		4 × 9 + 660 = 696.

Example VII.—Multiply 47 by 43:

47		43 and 7 are 50, which multiplied by the 4 of 47
43		makes 200, join an 0, equals	2000
2021		Multiply the unit 7 by 3, is	21
			2021

This mode may be better considered by putting letters for figures, 47 represented by z k, and 43 by r m.

The rule for these examples does not vary in principle from the first rule for figures under 20; for in this, the figure in the tens place of the multiplicand is used; in that it was unnecessary, for being a 1, to multiply by such number, could not increase it.

So far this rule can be applied with facility, when the figures in the tens places of the factors are alike; but when those figures are different, the process is not quite so simple; but a little practice will make it sufficiently ensy The rule is, to make the greater number the multiplier, and add it as before, to the right hand or unit figure of the multiplicand; then multiply such result by the tens figure of the muciplicand; the next step is to subtract the tens figure of the multiplicand, from the tens figure of the multiplier, then with this remainder multiply the unit figure of the multiplicand; such product, if a single figure, to be added to the last figure of the sum already had; but if such sum has three figures in it, and the product two, then the product will have to be put down in the units and tens places, and added in the common manner; to this last sum an 0 must be joined, after which, multiply the unit of the multiplicand, by the unit figure of the multiplier, and add such product to the former sum; being the true answer.

From reading this description it may appear a tedious plan, not worth the labour of acquiring a knowledge of it, but a few efforts will prove the contrary; and that a person without the aid of pen or paper, could work a sum much quicker than another with such aids.

Example I.—Multiply 24 by 36.

24		Add 36 to 4 makes 40, multiplied by the 2 of 24 is 80; as the difference between the 2 of 24 and the 3 of 36 is 1; it is one 4 of 24, which must be added to 80, making 84; Next join an 0 = 840, then multiply the 4 of 24 by the 6 of 36 is 24, added to 840 is 864.
36
864

		36 + 4 = 40
		40 × 2 = 80
The difference between 2 and 3 is 1		1 x 4 + 80 = 84 join an 0 = 840
		4 × 6 + 840 = 864

Example II.—Multiply 32 by 68.

32		Add 68 to 2, equals 70, which multiplied by 3 is 210; the difference between the 3 of 32 and the 6 of 68 being 3, is the multiplier of the 2 of 32, making 6, adding it to 210, is 216, to which join an 0, equals
68
2176

2160; then multiply the unit 2 by S, makes 16, added to 2160 equals 2176.

Example III.—Multiply 38 by 76.

38		Add 76 to 8 is 84, multiplied by 3 is 232, the difference between the 3 of 38 and the 7 of 76 is 4, by which figure multiply the 8 of 38, making 32, added to 252 is 284, join an 0, is 2840, next multiply 8 by 6 is 48, added to 2840 equals 2888.
76
2888

		76 + 8 = 84
		84 × 3 = 252
The difference between 3 and 7 is 4		4 x 8 + 252 = 284 join an 0 = 2840
		8 × 6 + 2840 = 2888

A different mode may be adopted, by making the lesser number the multiplier, and proceed as in this Example:—

42		Add 28 to 2 is 30, which multiplied by the 2 of 28 is 60, then subtracting the 2 of 28 from the 4 of 42 leaves 2, by which figure multiply 28, making 56, which added to 60 is 116, next join an O is 1160, multiply the units 8 by 2, is 16, plus 1160, equals 1176.
28
1176

Those two modes embrace all figures between 12 and 100, another arrangement is now submitted, which is in many instances superior.

Rule.—When the figures in the tens places are alike, and the figures in the units places by being added together, make 10; the figure in the tens place of the multiplicand must be increased 1; (which 1 ten is the sum of the units) then multiply them in the usual manner, putting down each product without any other combination.

Thus to multiply 27 by 23; the multiplicand 27 must be viewed as if it were 37.

Example I.

27	considered	37		Then say 3 times 7 are 21, which must be put down.—Twice 3 are 6, prefixed to 21 is 624.
23	---	23
621	---	621

Example II.—46 times 44.

46	considered as	56		6 × 4 = 24 which put down. 5 × 4 = 20 prefixed to 24 = 2024.
44	---	44
2024	---	2024

When the figures in the units places, by being added together make more than 10, the excess must be noted; and after the units have been multiplied, the figure in the tens place of the multiplier, must be multiplied by the excess alluded to, which sum must be added to the tens figure gained by the multiplication of the units, afterwards proceed as in the former example.

Example I.—Thus 27 times 24 must be viewed as 37 times 24.

27	considered	37		4 added to 7 makes 11, which is 1, above 10; this figure must be used afterwards; for the addition of the units is not necessary to work the sum, being only requisite to ascertain the excess of 10.
24	---	24
648	---	648

The 7 of 37 must be multiplied by the 4 of 24, making 28, the 8 is to be put down as part of the product; next multiply the 2 of 24 by the excess 1, making 2, which is to be added to the 2 of 28, making 4; joined to the 8 is 48: then multiply the figures in the tens places, 3 by 2 gives 6, joined to 48 is 648.

27	multiplied as	37		7 × 4 = 28 put down 8 and carry 2
24	---	24		2 x 1 + 2 = 4 put down 4
648	---	648		2 × 3 = 6 joined to 4, and 8 = 648.

To work the sum with the letters that represent the figures, may make it less liable to mistake; we shall call 37, m k, and 24, h r.

m k		4 and 7 are 11, being 1 above ten, call such excess q.
h r
d z b

r × k = 28 which call n b
h × q + n = 4 call z
h × m = 6 call d
Join d, z, b together, is the sum = 648.

Example II.—48 multiplied by 45,—view 48 as 58.

48	viewed as	58
45	---	45
2160	---	2160

As 5 and 8 are 13, the excess of ten is 3. Multiply 8 by 5 is 40, put down 0, and carry 4, next multiply the 4 of 45 by the excess 3, equals 12; added to the 4 that was carried makes 16, put down 6 before the 0 of 40, and carry 1; then multiply the 5 of 58 by the 4 of 45 makes 20, added to the 1 that was carried makes 21, which prefixed to the 60 already had, is 2160; the correct sum of 48 multiplied by 45.

When the figures in the tens places are not alike, and the unit figures by being added together make 10, act as in this example—58 times 32.

58
32
1856

8 multiplied by 2 makes 16, put down 6 and carry 1, then subtract the 3 of 32 from the 5 of 58, leaves 2, with this figure, multiply the 2 of 32, making 4, which added to the 1, that was carried from 16, makes 5; which must be put down before the 6, then considering the 5 of 58, as a 6, according to former examples; multiply it by the 3 of 32 makes 18, put down before 56, gives the product 1856.

These examples are purely given, in hopes that the faint liglıt which is thrown upon this mode of multiplying figures, may induce some person that has leisure, to devise a more complete method, by making (if possible) one general rule, for such or similar calculations. The same motive induces the writer to give an example or two, of some cases where three figures may be multiplied in the mind.

Rule.—When the figures in the tens and the units places, are alike in the multiplier and in the multiplicand, and the unit figures, by being added together, make 10-proceed like the first examples.

Example—136 multiplied by 134.

136
134
18224

Add 134 to 6 makes 140; reject the O and consider the sumas 14, with which multiply the 13 of 136, first adding 14 to the 3 of 13, makes 17; join to it an 0, cquals 170, then the 4 of 14 and the 3 of 13, being multiplied by each other, gives 12, added to 170 is 182—then multiply the unit 6 by the unit 4 gives 24—joined to the 182 already had gives the correct sum.

Example II.—262 by 268.

262		268 + 2 = 270, reject 0, leaves 27 027 + 6 = 33 033 x 2 = 66, join 0 = 660 006 × 7 + 660 = 702 002 × 8 = 16, joined to 702 = 70216.
268
70216

Another mode of working similar sums is conformable to the examples, page 132.

Multiply 147 by 143.

147
143
21021

Multiply the 7 of 147 by 3, equals 21, which put down as a part of the product, then increasing the 4 in the tens place of the multiplicand a 1, makes 5, (the tens figure of the multiplicand must always be increased 1), which multiplied by the 4 of the multi-plier, makes 20, put down 0 and carry 2, then add it to the tens figures of the multiplier and the multiplicand, and then to the figure in the hundreds place of the multiplicand; in this instance, say 2 and 4 are 6, and 4 are 10, and 1 are 11, put down 1, and carry 1, to the 1 of 143 makes 2, prefix it to the other figures, gives 21021.

147			Q r k		r made k × m = 21 r made 5 × z = 20 calls n s r made n + z + r + Q = 11 call q' t r made T + q' = 2 call h Join h t s to 21 = 21021
143			T z m
21021			h t s n t

Many other examples of a similar nature might be given, these will suffice to shew the outlines of the prominent ones; but we shall conclude this chapter by another method which although not new, yet as it is not generally known may be of service. This mode may appear complicated but a little practice will make it easy.

The letters that represent the figures mnemonically, will be put under the figures of the multiplier and the multiplicand, and will be so continued throughout the operation.

Example—Multiply 234 by 512 in one line.

2	3	4	multiplicand
h	m	r
5	1	2	multiplier.
l	t	n
119808

2 × 4 = 8 put down 8 as part of the product.
n × r
2 × 3 = 6 call d
n × m
1 × 4 + d = 10 put down 0 and carry 1, call q
t × r
2 × 2 + q = 5 call j
n × h
5 × 4 + j = 25
l × r
1 × 3 + 25 = 28 put down 8 and carry 2, call h'
t × m
1 × 2 + h' = 4 call z
t × h
5 × 3 + z = 19 put down 9 and carry 1, call q'
l × m
5 × 9 + q' = 11 put down.
l × n

Which in words would be as follows:

Twice 4 are 8, put down 8.

Twice 3 are 6 and (once 4) 4 are 10, put down 0 and carry 1.

Twice 2 are 4 and 1 are 5, and (5 times 4) 20 are 25, and (3 times 1) 3 are 28, put down 8 and carry 2.

Once 2 are 2, and 2 are 4, and (5 times 3) 15, are 19, put down 9 and carry 1; 5 times 2 are 10, and 1 are 11, which put down.

This method may be extended to any number of figures; the plan consists in first multiplying the two figures that are in a straight line, or opposite each other; then in a diagonal line from the first figure of the multiplier to the second figure of the multiplicand, next diagonally from the second figure of the multiplier to the first figure of the multiplicand, and in the same manner through the whole sum.

Another example is given of multiplying four figures by four figures, which need not be put down algebraically, for the knowledge of the method, by which the former sum was effected, will direct in this.

4653

7428

34562484

8 times 3 are 24, put down 4 and carry 2.

8 times 5 are 40 and (2 carried) 2, are 42, and (2 × 3) 6, are 48; put down 8 and carry 4.

8 times 6 are 48 and (4 carried) 4 are 52, and (4 × 3) 12, are 64 and (2 × 5) 10 are 74, put down 4 and carry 7.

8 times 4 are 32 and (7 carried) 7 are 39, and (7 + 3) 21, are 60, and (2 × 6) 12 are 72 and (4 × 5) 20 are 92; put down 2 and carry 9.

Twice 4 are 8, and (9 carried) 9 are 17, and (7 × 5) 35 are 52, and (4 x 6) 24 are 76, put down 6 and carry 7.

4 times 4 are 16 and (7 carried) 7 are 23, and (7 x 6) 42, are 65, put down 5 and carry 6.

7 times 4 are 28, and (6 carried) 6 are 34.

When there are fewer figures in the multiplier than in the multiplicand, it will be much easier to work the sum, than when they are equal in numbers, as this example will shew.

5321

62

329902

Twice 1 are 2, which put down.

Twice 2 are 4 and (6 × 1) 6 are 10, put down 0 and carry 1:

Twice 3 are 6 and (1 carried) 1 are 7, and (6 × 2) 12 are 19 put down 9 and carry 1.

Twice 5 are 10, and (1 carried) 1 are 11, and (6 × 3) 18 are 29, put down 9 and carry 2.

6 times 5 are 30 and (2 carried) 2 are 32.