Question

Wei Jing has between 70 and 150 baseball

cards in his collection. When he arranges them
in groups of 10, he has 3 left over. When he
arranges them in groups of 9, he has 5 left
over. If he arranges them in groups of 8, how
many will he have left over?

Answer 1

117$.

Explanation:

Let $x$ represent the number of cards in Wei Jing's collection. We are given that $70 \leq x \leq 150$. We are also given that $x \equiv 3 \pmod{10}$ and $x \equiv 5 \pmod{9}$. Since $x \equiv 3 \pmod{10}$, $x - 3$ is divisible by 10. Since $x \equiv 5 \pmod{9}$, $x - 5$ is divisible by 9. The least common multiple of 10 and 9 is $90$, so $(x - 3) - 90$ is divisible by 90 and $(x - 5) - 90$ is divisible by 90. Adding these equations gives us $2x - 98$ is divisible by 90. Dividing both sides by 2 gives us $x - 49$ is divisible by 45. Since $70 \leq x \leq 150$, the only solution for $x$ is $x = 117$.