Divisibility of numbers: divisibility rules

Practice:

👉 Exercises on divisibility rules. 👈

What is divisibility

A number a is said to be divisible by a number b without a remainder if there exists a number c such that a = b · c. It is denoted: a ⋮ b.

Divisibility rules allow you to quickly determine whether a number is divisible by another without performing the division itself.

Rules for 2, 3, 5, 9, 10

Main divisibility rules:

By 2 — if the last digit is even: 0, 2, 4, 6, 8;
By 3 — if the sum of digits is divisible by 3;
By 5 — if the last digit is 0 or 5;
By 9 — if the sum of digits is divisible by 9;
By 10 — if the last digit is 0.

Rules for 4, 6, 8

Additional rules:

By 4 — if the last two digits form a number divisible by 4;
By 6 — if the number is divisible by both 2 and 3;
By 8 — if the last three digits form a number divisible by 8.

Rules for 7, 11

Less obvious but useful rules:

By 7 — separate the last three digits, subtract the smaller number from the larger one; if the difference is divisible by 7, then the original number is divisible too;
By 11 — the difference between the sum of digits in odd positions and the sum of digits in even positions must be divisible by 11.

Examples

Examples of applying divisibility rules:

124 is divisible by 2 (the last digit is 4 — even);
123 is divisible by 3 (1 + 2 + 3 = 6, divisible by 3);
125 is divisible by 5 (the last digit is 5);
243 is divisible by 9 (2 + 4 + 3 = 9);
1316 is divisible by 4 (the last two digits 16 are divisible by 4);
126 is divisible by 6 (divisible by both 2 and 3).

Frequently asked questions

What is a divisibility rule?

A divisibility rule is a rule by which you can quickly determine whether one number is divisible by another without performing the division. For example, a number is divisible by 2 if its last digit is even.

How do you check divisibility by 3?

You need to add all digits of the number and check whether the sum is divisible by 3. If yes, the number is divisible by 3. For example, 123: 1 + 2 + 3 = 6, 6 is divisible by 3, so 123 is divisible by 3.

Which divisibility rules do you need to know for the EGE?

For the EGE and OGE, you basically need to know the divisibility rules for 2, 3, 4, 5, 6, 8, 9, 10. The rules for 7 and 11 occur less often but are also useful.

Is 0 divisible by any number?

Yes, 0 is divisible by any non-zero number, and the result is always 0. However, division by 0 is undefined.