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Multiplication by 8 and 7

In last week’s issue of this newsletter, we discussed a quick way of multiplying by 9. We will extend that method to multiplying by 8 and 7. As we have pointed out in our book, Algebra Made Easy as Arithmetic, this is not the recommended technique in arithmetic, but it is useful in polynomial multiplication when one factor is (x – 2), (x – 3) or, (x – a) where a is any positive integer.

If x = 10, the (x – 2) = 8 and (x – 3) = 7.

Consider 124 x 8:

134 x 8 = 134 x (10 – 2) = 1340 – 268.

In the traditional way, the subtraction is:

    1 3 4 0

  •  2 6 8

    1 1(2)(8) = 1072

Here we see that 2, which is twice 1, the first digit of the multiplicand is subtracted from 3, which is the next digit to the right of 1.

Next, we see that 6 which is twice 3, the second digit of the multiplicand is subtracted from 4 which is next to 3.

Lastly, 8, which is twice 4, the last digit of the multiplicand is deducted from 0 which was added at the end. We can just place the negative of twice the last digit of the multiplicand in bar form as the last digit of the answer.

Now we can state the rule for multiplying by 8.

  1. Copy the first digit of the multiplicand as the first digit of the product.
  2. Starting from the first to the penultimate digit, subtract twice each digit from the next one to the right. Write down any negative difference in bar form.
  3. Write the negative of twice the last digit of the multiplicand as the last digit of the answer.
  4. Rewrite the answer in normal format.

Thus, we have,

1342 x 8 = 11(2)(6)(4) = 10,736

2475 x 8 = 20(1)(9)(10) = 20(1)(100) =  20(200) = 18,000

 5846 x 8 = 5(2)(12)(2)(12) = 5(2)(12)(32) = 5(2)(1232) = 5(3232) = 45,768

Similarly, when multiplying by 7 we subtract thrice from the next digit.

123 x7 = 1(1)(3)(9) = 861

1348 x 7 = 10(5)(4)(24) = 10(5)(64) + 10(564) = 9436

1384 x 7 = 10(1)(20)(12) = 10(1)(212) = 10(312) = 9688  

Try these multiplications:

  1. 35 x 8 =
  2. 35 x 7 =
  3. 53 x 8 =
  4. 53 x 7 =
  5. 251 x 8 =
  6. 251 x 7 =
  7. 4253 x 8 =
  8. 4253 x 7 =
  9. 7643 x 8 =
  10. 7643 x 7 =

Answers to previous week’s exercises

  1. 35 x 9 = 32(5) = 315
  2. 53 x 9 = 5(2)(3) = 477
  3. 258 x 9 = 233(8) = 2322
  4. 285 x 9 = 26(3)(5) = 2565
  5. 3478 x 9 = 3131(8) = 31,302
  6. 3749 x 9 = 34(3)5(9) = 33,741
  7. 25688 x 9 = 23120(8) = 231,192
  8. 52368 x 9 = 5(3)132(8) = 471,312
  9. 364869 x 9 = 33(2)4(2)3(9) = 3,283,821
  10. 754868 x 9 = 7(2)(1)4(2)2(8) = 6,793,812
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