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Problem #7 · August 3, 2026

EKUB greatest common divisor and EKUK least common multiple

The greatest common divisor of two natural numbers equals 6, and their sum equals 84. How many such pairs of numbers are there? (The order of the numbers in the pair is not taken into account.)
  1. A3
  2. B4
  3. C6
  4. D7
  5. E13

WORKED EXAMPLE

Solution

We write the numbers as 6m6m and 6n6n, where EKUB(m;n)=1EKUB(m; n) = 1 must hold. Then 6(m+n)=846(m + n) = 84, that is, m+n=14m + n = 14. Coprime pairs m<nm < n: (1;13),(3;11),(5;9)(1; 13), (3; 11), (5; 9) — (7;7)(7; 7) does not fit, because EKUB(7;7)=7EKUB(7; 7) = 7. Therefore the pairs are (6;78),(18;66),(30;54)(6; 78), (18; 66), (30; 54) — 3 in total.
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