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

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.

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

In determining divisibility by any number, two facts must be remembered:

- Addition of zeroes to and subtraction of zeroes from the end of a number does not affect its divisibility by a divisor.
- Addition or subtraction of the divisor or a multiple of a divisor does not affect the divisibility of that number.

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. Thus, 2,346 is not divisible by 4.

Common divisibility tests will be discussed by IAVM Trustee Gowri Ramachandran on Dec 5 while Osculation, a systematic divisibility test for any number will be explained by MATH-Inic Training Director Veronica S. Prudente on Dec 11, 2021, in the Inspirational Maths from India (Year 4) webinars. (Please send a message to “Ike Prudente” on Messenger or to https://www.facebook.com/MATHInicPhils for registration details.)

Is 8,733 divisible by 41? We subtracted multiples of 41 repeatedly from 8,733 until we are left with 0. Obviously, 8,733 is divisible by 41.

We can easily create zeroes at the end of the dividend by multiplying the last digit of the number, in this case 3, by 41 and subtracting the product, 123, from the original number.

8,733 – 3 x 41 = 8,733 – 123 = 8,610

We can then remove the zero at the end and repeat the process.

861 – 1 x 41 = 861 – 41 = 820.

Now many can recognize that 820 is divisible by 41.

We can repeat the process until we arrive at a more conclusive result.

82 – 2 x 41 = 82 – 82 = 0

A further modification of this method is to use 4, not 41: Remove the last digit of the dividend and multiply it by 4. Then subtract the product from the resulting number.

For 8,733, we have:

873 – 3 x 4= 873 – 12 = 861

86 – 1 x 4 = 82

8 – 2 x 4 = 0

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