Multiplication by 9

Upper elementary grade learners can easily multiply by 9 using the conventional right to left method. But there are simpler ways to know the correct answer.

We know that a number is divisible by 9 if the sum of it digits is divisible by 9

Since one of the multiplicands is 9, it follows that the product must be divisible by 9. Now by using digit sums, we can immediately eliminate the choices with having digit sums not divisible by 9: a) 111211 → 7; b) 112211  → 8 and e) 130121 →8. Both choices c and d have a digit sum of 9. But which one is correct?

In our book “Algebra Made Easy as Arithmetic”( available at https://www.facebook.com/MATHInicPhils), in addition to an extensive discussion of the applications of digit sums, an alternative way to multiply by 9 is presented.

If we express 9 as (10 – 1) we can mentally multiply short numbers by 9 using subtraction.

Thus, we have:

            7 x 9 = 7 x (10 – 1) = 70 – 7 = 63

            57 x 9 = 570 – 57 = 513

            248 x 9 = 2480 – 248 = 2232

We can develop a further technique in multiplying by 9. Let us use the last example above.

           2 4 8 0

               2 4 8

            2 2 4 (8) → 2240 – 8 →  2 2 3 2

Note: the digit enclosed in parentheses is negative

We can see that starting from the left each digit is subtracted from its neighbor on the right while the last digit is subtracted from 0 resulting to its additive invers or negative.

For our featured example, we can simply the computation to:

12479 x 9  → 1| 2 – 1| 4 – 2 | 7 – 4 | 9 – 7| 0 – 9 → 11232(9) → 112311

We will discuss more complicated examples next week

Practice exercises:

1. 36 x 9 =
2. 28 x 9 =
3. 123 x 9 =
4. 136 x 9 =
5. 258 x 9 =
6. 236 x 9 =
7. 1456 x 9 =
8. 2569 x 9 =
9. 12378 x 9 =
10. 14679 x 9 =

Answers to last week’s exercises:

1. 28 – divisible
2. 34 – not divisible
3. 38 – not divisible
4. 44 – divisible
5. 52 – divisible
6. 58 – not divisible
7. 68 – divisible
8. 76 – divisible
9. 86 – not divisible
10. 94 – not divisible