The number that is one short of every multiple

Question

What is the smallest positive whole number that leaves a remainder of 1 when divided by 2, a remainder of 2 when divided by 3, a remainder of 3 when divided by 4, and so on, up to a remainder of 9 when divided by 10?

Accepted Answer

2,519. Reframe. "Remainder 1 on division by 2" means the number is one less than a multiple of 2. "Remainder 2 on division by 3" means it is one less than a multiple of 3. In every case, the number is one less than a multiple of the divisor. So the number plus one is a multiple of 2, 3, 4, 5, 6, 7, 8, 9 and 10 all at once. The least common multiple. The smallest positive number divisible by all of 2 through 10 is their least common multiple: take the highest power of each prime that appears, 2 cubed (from 8), 3 squared (from 9), 5 and 7. That is 8 × 9 × 5 × 7 = 2,520. Answer. The number is 2,…

The number that is one short of every multiple

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