Elementary arithmetic, with brief notices of its history... by Robert Potts.

19
Let it be required to find the product of 5342 and 324.
The multiplier 324 = 300 + 20 + 4.
If the multiplicand be first taken 4 times, then 20 times, and thirdly
300 times, the sum of these three partial products will be equal to
324 times 5342, or the product required.
The process may be thus exhibited:
5342
324 = 300 + 20 + 4.
21368 = 4 times 5342.
106840 = 20,,
1602600 = 300,
1730808 = 324,,
In practice it is usual to omit the ciphers, taking care to place the
first figure of each partial product under the figure which forms the
multiplier for that product.'
The product of two numbers can also be found by multiplying one
of them by the factors into which the other can be divided.
Thus the product of 5342 by 324 can be found by multiplying
5342 by 4, 9, 9 in succession, the factors of 324.
5342
4
21368 - 4 times 5342.
9
192312 = 36 times 5342.
9
1730808 = 324 times 5342.
1 The following is the proof of the process of multiplication "by casting out the
nines." It depends on the property that "any number divided by nine will leave
the same remainder as the sum of its digits divided by nine."
First. Cast out the nines from the sums of the digits of the multiplicand and of
the multiplier, and reserve the remainders.
Secondly. Multiply these two remainders together and cast out the nines from
their product, and reserve the remainder.
Thirdly. Cast out the nines from the sum of the digits of the product, and
reserve the remainder.
Fourthly. Then if these two remainders be equal, the process of the multiplication is correct.
This so-called proof is defective as a proof in the following particulars, as it fails
to detect errors in the product1. If the order of figures in the product be misplaced, as 37 for 73.
2. If errors be made which counterbalance each other, as 35 written for 62, the
sums of the digits in each case being the same.
If 9 be written for 0, or 0 for 9, or either be omitted or inserted too often.
The proof of multiplication "by casting out the elevens" is not liable to so
many chances of failure as the proof "by casting out the nines."