Divisibility Intuition Math Example 2

Follow the full solution, then compare it with the other examples linked below.

medium

Explain why the divisibility rule for

3

works: a number is divisible by

3

if and only if the sum of its digits is divisible by

3

1
Any number can be written in terms of its digits. For a three-digit number: $\overline{abc} = 100a + 10b + c$ .
2
Rewrite: $100a + 10b + c = (99a + 9b) + (a + b + c) = 9(11a + b) + (a+b+c)$ .
3
The term $9(11a+b)$ is always divisible by $9$ (and hence by $3$ ).
4
So $\overline{abc}$ is divisible by $3$ if and only if $a+b+c$ is divisible by $3$ . The same argument extends to any number of digits.

Answer

The rule works because

10^k \equiv 1 \pmod{3}

for all

k

, so the number's remainder mod

3

equals the digit sum's remainder mod

3

The divisibility-by-

3

rule is not magic: it follows from the fact that

10 = 9+1 \equiv 1 \pmod 3

, so each place value contributes the same as the digit itself. Understanding why rules work is more powerful than memorising them.

About Divisibility Intuition

Understanding when one whole number divides evenly into another, leaving no remainder—the foundation of factor and multiple relationships.

Use divisibility rules to determine whether [formula] is divisible by [formula], [formula], [formula

Test [formula] for divisibility by [formula], [formula], and [formula] using divisibility rules.

A number [formula] leaves remainder [formula] when divided by [formula] and remainder [formula] when