 # MSC #15 – Divisibility: By Addition and by Subtraction

Our 15th MATH-Inic Special for Christmas deals with divisibility of numbers.

Grade school children are required to know by heart the divisibility rules for numbers such as 2, 3, 4, 5, 7,  8, 9, 10 and 11.  Also a number is divisible by a composite number, if it is divisible by all its factors.

A non-conventional, but logical method, to determine the divisibility of a number is by addition or subtraction of any multiple of the divisor.

Is 1393 divisible by 7? By just adding 7 to 1393 we get 1,400 which obviously is divisible by 7. So 1393 is divisible by 7.

If 2346 divisible by 4? Since 100 is divisible by 4, we can deduct 2300 (23 x100) from 2346 to get 46. Then we can subtract 40 from 46 and get 6 which is obviously not divisible by 4. So 2,346 is not divisible by 4.

Is 47,492 divisible by 31?  We subtracted multiples of 31 repeatedly from 47,492 until we are left with 31,000, or 31 x1000, a number that we can easily identify as divisible by 31.

Since the divisor (31) ends in 1, we can easily create zeroes at the end of the dividend. Subtract the product of 31 and the last non-zero digit of the dividend (or resulting dividend). Eliminate the zero at the end. We repeat the process until we are left with a number that we can mentally classify as divisible or not divisible by 31.

47, 492 – 2 x 31 = 47430

4743 – 3 x 31 = 4650

465 –  5 x 31 = 310

Could you see the similarity between the method we just used and the divisibility test for 7?

This “By addition and by Subtraction” method of determining the divisibility  of a number will be fully discussed in chapter 15 of our coming book “30 Master Strategies in Computing”