Problem #37 · September 2, 2026

Divisibility rules

How many three-digit numbers of the form

\overline{aba}

are divisible by

4

WORKED EXAMPLE

Solution

For a number to be divisible by

4

, the number

\overline{ba} = 10b+a

formed by its last two digits must be divisible by

4

; divisibility of

10b+a

and

2b+a

4

is equivalent, so in particular

a

must be even. If

a \in \{4; 8\}

b

must be even —

2 \cdot 5 = 10

numbers; if

a \in \{2; 6\}

b

must be odd — another

2 \cdot 5 = 10

numbers. In total,

10+10=20

numbers (for example,

424

616

848