Problem 27 · 2023 Math Kangaroo
Stretch
Number Theory
divisibilityfactorization
What is the biggest common factor of all numbers of the form \(n^3 (n+1)^3 (n+2)^3 (n+3)^3 (n+4)^3\) where n is a positive integer?
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Answer: E — \(2^9 \cdot 3^3 \cdot 5^3\)
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Hint 1 of 2
Inside the cube, the base is a product of five consecutive integers.
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Hint 2 of 2
Find the gcd of 'product of five consecutive integers' over all n, then cube it.
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Approach: find the gcd of five-consecutive-integer products, then cube
- A product of five consecutive integers is always divisible by 5! = 120, and n = 1 gives exactly 120, so the gcd of these products is 120 = 23·3·5.
- The given numbers are that product cubed, so their gcd is 1203 = 29·33·53.
- That is option 29·33·53.
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