Testing for Divisibility

Among numerous other factors, good scores on any standardized test such as GRE, SAT or GMAT depends on the student’s ability to perform the calculations faster.

Divisibility checking can be time consuming at times. So, any trick to save a minute or two can help a student significantly.

For illustration purposes I am going to consider numbers of form:

 abcd

Where:

-          d is in unit’s place,

-          c is in ten’s place,

-          b is in hundred’s place,

-          a is in thousand’s place.

So, if I had a number 8976 then d = 6, c = 7, b = 9, a = 8.

Of course, the concept I talk about here using an example of 4 digit numbers can be applied to 2 digits, 3 digits, 5 digits, 10 digits, etc. numbers as well.

Divisibility by 2:

-           A number is divisible by 2 ONLY IF the unit’s place of the number is divisible by 2.

-          i.e. d = 0, 2, 4, 6, or 8.

-          E.g. 2453 is not divisible by 2, but 2454 is divisible by 2.

Divisibility by 3:

-           A number is divisible by 3 ONLY IF the sum of the digits of the number is divisible by 3.

-          i.e. (a + b + c + d) has to be divisible by 3.

-          E.g. 2453 is not divisible by 3 as 14 (= 2 + 4 + 5 + 3) is not divisible by 3.

-          However, 2451 is divisible by 3 as 12 (= 2 + 4 + 5 + 1) is divisible by 3.

-          In the same way, a number is divisible by 9, 27, 81, etc. if the sum of digits is divisible by 9, 27, 81, etc. RESPECTIVELY.

Divisibility by 4:

-           A number is divisible by 4 ONLY IF the number formed by the units and ten’s place number is divisible by 4.

-          i.e. the number cd has to be divisible by 4.

-          E.g. 2453 is not divisible by 4 as 53 is not divisible by 4.

-          However, 2452 is divisible by 4 as 52 is divisible by 4.

Divisibility by 5, 25:

-           A number is divisible by 5 ONLY IF the unit’s place of the number is divisible by 5.

-          i.e. d = 0 or 5.

-          E.g. 2453 is not divisible by 5 but 2455 is divisible by 5.

-          Similarly a number will be divisible by 25 if cd is divisible by 25.

Divisibility by 6, 12, etc.:

-           A number is divisible by 6 ONLY IF it is divisible by 2 and 3 simultaneously. This works because 2 and 3 are factors of 6.

-          Similarly a number will be divisible by e.g. 12 if it is divisible by 3 and 4 (12 = 3 x 4).

Divisibility by 10:

-           A number is divisible by 10 ONLY IF d = 0.

Divisibility by 11:

-           A number is divisible by 11 ONLY IF the difference of the sum of alternate numbers is divisible by 11.

-          i.e. (a + c) – (b + d) is divisible by 11.

-          E.g. 2452 is not divisible by 11 as 1 ([2 + 5] – [4 + 2]) is not divisible by 11.

-          However, 2453 is divisible by 11 as 0 ([2 + 5] – [4 + 3]) is divisible by 11.

The consequence of the above tricks is that

-          All the remainders would be zero (0) the moment the divisibility condition is satisfied.

-          There is no need to waste time punching a calculator!

There are divisibility tricks for other numbers as well but they are not as popular as those I have described here.