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Problem 363

Number theory Difficulty 2.5 Multiple choice CEMC Pascal · Canada · 2023

Nasim buys trading cards in packages of 5 cards and in packages
of 8 cards. He can purchase exactly 18 cards by buying two 5-packs and
one 8-pack, but he cannot purchase exactly 12 cards with any combination
of packages. For how many of the integers $n=24, 25, 26, 27, 28, 29$ can he buy
exactly $n$ cards?

Pick one

Official solution

Nasim can buy 24 cards by buying three 8-packs ( $3 \times 8 = 24$ ).

Nasim can buy 25 cards by buying five 5-packs ( $5 \times 5 = 25$ ).

Nasim can buy 26 cards by buying two 5-packs and two 8-packs ( $2 \times 5 + 2 \times 8 = 26$ ).

Nasim can buy 28 cards by buying four 5-packs and one 8-pack ( $4 \times 5 + 1 \times 8 = 28$ ).

Nasim can buy 29 cards by buying one 5-pack and three 8-packs ( $1 \times 5 + 3 \times 8 = 29$ ).

Nasim cannot buy exactly 27 cards, because the number of cards in
8-packs that he buys would be 0, 8, 16, or 24, leaving 27, 19, 11, or 3
cards to buy in 5-packs. None of these are possible, since none of 27,
19, 11, or 3 is a multiple of 5.

Therefore, for 5 of the 6 values of $n$ , Nasim can buy exactly $n$ cards.

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.