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Problem #10 · August 6, 2026

Divisibility rules

If the four-digit number 53a6‾\overline{53a6} is divisible by 44, how many different values can the digit aa take?
  1. A3
  2. B4
  3. C5
  4. D6
  5. E10

WORKED EXAMPLE

Solution

Divisibility by 44 depends on the number a6‾=10a+6\overline{a6} = 10a+6 formed by the last two digits. If we check, the numbers 1616, 3636, 5656, 7676, 9696 are divisible by 44, whereas for even aa the numbers 0606, 2626, 4646, 6666, 8686 are not divisible. Therefore aa must be odd: a∈{1;3;5;7;9}a \in \{1; 3; 5; 7; 9\} — 55 values in total.
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