Problem 5 · 2016 AMC 8
Medium
Number Theory
divisibilitymod-10
The number N is a two-digit number.
- When N is divided by 9, the remainder is 1.
- When N is divided by 10, the remainder is 3.
What is the remainder when N is divided by 11?
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Answer: E — Remainder 7.
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Hint 1 of 3
The cheapest clue to use is the ÷10 one: a leftover of 3 when you divide by 10 simply means N ends in the digit 3. That single fact shrinks the whole list to {13, 23, 33, …, 93}.
Still stuck? Show hint 2 →
Hint 2 of 3
Now apply the ÷9 clue. Handy shortcut: a number's leftover when divided by 9 equals the leftover of its DIGIT SUM — so just look for the candidate whose digits add to something 1-more-than-a-multiple-of-9.
Still stuck? Show hint 3 →
Hint 3 of 3
Find N first; only THEN divide it by 11. Don't try to juggle all three divisors at once — pin down the number, then answer the actual question.
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Approach: let the easy clue filter the list, then test with the digit-sum trick
- "Leftover 3 when divided by 10" means the units digit is 3, so N ∈ {13, 23, 33, 43, 53, 63, 73, 83, 93}.
- Digit-sum test for division by 9 (a number and its digit sum leave the same leftover): 13→4, 23→5, …, 73→10 which leaves 1. Only 73 leaves a leftover of 1. So N = 73.
- Now answer the real question: 73 = 11 × 6 + 7, so the leftover when divided by 11 is 7.
- Why this transfers: when several remainder conditions overlap, start with the one that fixes the most (divide-by-10 fixes a whole digit), then sieve the short list — far faster than algebra.
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