Problem 13 · 2012 Math Kangaroo
Hard
Number Theory
divisibilityfactorization
When the numbers 144 and 220 are divided by the same positive whole number x, both have remainder 11. Find x.
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Answer: D — 19
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Hint 1 of 2
If x leaves remainder 11, then x divides each number minus 11.
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Hint 2 of 2
Find a common divisor of 144−11 and 220−11 that is bigger than 11.
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Approach: subtract the remainder, then take a common divisor
- Since both leave remainder 11, x divides 144 − 11 = 133 and 220 − 11 = 209.
- Factor: 133 = 7 × 19 and 209 = 11 × 19, so their common divisor bigger than 11 is 19.
- x must exceed the remainder 11, so x = 19 (D).
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