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Problem #35 · August 31, 2026

Divisibility rules

The four-digit number 4a1b‾\overline{4a1b} is divisible by 4545. Find the greatest value of the sum a+ba+b.
  1. A4
  2. B8
  3. C9
  4. D12
  5. E13

WORKED EXAMPLE

Solution

Since 45=9⋅545 = 9 \cdot 5, the number must be divisible by both 99 and 55, so b∈{0;5}b \in \{0; 5\}. When b=0b=0, the digit sum is 5+a5+a, giving a=4a=4; when b=5b=5, the digit sum is 10+a10+a, giving a=8a=8. The sums are 4+0=44+0=4 and 8+5=138+5=13, and the greatest is 1313 (check: 4815=45⋅1074815 = 45 \cdot 107).
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