Divisibility by 7

Most Grade 3 or Grade 4 learners are taught a method of determining a number’s divisibility by 7. In 2019, a 12-year-old Nigerian boy Chika Ofili, became an internet sensation when he received a special award from his school because he “discovered” a new formula for divisibility by 7.

We will discuss those two methods in the following weeks but for now we will discuss a procedure which we used in determining divisibility by 4 to show that it will work for all divisors. Also, we will illustrate how a special technique can make this procedure easy to use.

In explaining divisibility by 4, we used the Vedic Mathematics Sutra or word formula “By Addition and By Subtraction”. Adding or subtracting a divisor or multiple of a divisor to a number does not affect its divisibility by that divisor. Also, addition or subtraction of zeroes to or from the end of a number does not affect its divisibility by a divisor.

 By Addition and by Subtraction is like the long division except that it can be done also from right to left. By combining it with a technique called number splitting, we can sometimes easily spot numbers divisible by a certain divisor.

In option a) 14422884 of our problem we can deduct successively, multiples of 7 to reduce it into a small number which we can immediately identify as either divisible or not divisible by 7. This can be done as follows:

            14422884

          –   14000000

       422884

               –   420000

                        2884

                    –   2884   

                              84

This can also be done from right to left, deducting 84 first to get 14422800 and then disregarding the two ending zeroes.

By using number splitting we can express it as 14 | 42 | 28 | 84 which is equivalent to decomposing the number into 14,000,000 + 420,000 + 2,800 + 84. In this way we can determine the quotient: dividing each partition by 7 will give us 02 | 06 | 04 | 12, therefore 14,422,884 ÷ 7 = 2,060,412.

For option b) 22444884, we can not partition it into four groups of 2 digits each since 22 is not divisible by 7. However, it is easy to see that 224 is 210 + 14 and the next three digits 448 is double 224. So, we can split it into 224 | 448 | 84 which when divided by 7 will give 32 | 064 | 12 or 3,206,412.

We will use a different technique for option c) 40684672. If we add 28 to it, we will 40684700. We can either 1) disregard the two ending zeroes giving 406847, then we subtract 7 to yield 406840 and again disregard the ending zero to give 40684 or 2) immediately disregard the ending 700 which is a multiple of 7. We can then remove 84 to get 406. Then By addition of 14, we will get 420 which is clearly divisible by 7.

For option d) 42294364 we will use a combination of the previous methods. First, we can disregard the initial two digits 42 because it is divisible by 7 leaving use with 294364. From this we subtract 14 to get a remainder of 294350. We can then disregard the ending 350, which leaves us with 294. If we subtract 14 from it, we will get 280 which is clearly divisible by 7.

For choice e) 46222444, while we cannot see a two-digit multiple of 7, we can recognize the initial digits 462 as 420 + 42 and the next three digits as 224 as 210 + 14. Therefore, we can partition the number into 462 | 224 |44. When we divide this by 7, we will get 66 |032 | 06 rem 2.

Practice Exercises:

Indicate if the number is divisible or not divisible by 7

1. 763 →
2. 392 →
3. 1444 →
4. 4445 →
5. 23331 →
6. 38885 →
7. 424807 →
8. 133651 →
9. 7638476 →
10. 8456728 →

Answers to previous exercises:

State whether the number is divisible by 18 or not

1. 91351351351359 → not divisible
2. 32342342342346 → divisible
3. 25225225252252 → divisible
4. 40540540540545 → not divisible
5. 77220022777722 → divisible
6. 39999999999963 → not divisible
7. 56756799567567 → not divisible
8. 33366666633363 → not divisible
9. 12345678234567 → not divisible
10. 43234323432345 → not divisible