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Perfect powers, divisibility and constrained prime factors
Free examples and worked answers. No account needed.
Practise the method.
Perfect powers, divisibility and constrained prime factors
Perfect powers, divisibility and constrained prime factors
Example 1
N=711×1712. Find the smallest positive integer m for which mN is a perfect square.
Show worked answer
Answer: 7
- A square has even prime exponents.
- Supply one additional copy of each prime whose exponent is odd: m=7.
Example 2
N=711×1714. Find the smallest positive integer m for which mN is an integer perfect cube.
Show worked answer
Answer: 14161
- Remove 72×172 so both remaining exponents are multiples of 3.
- The smallest divisor is 14161.
Example 3
N=313×1310. Find the smallest positive integer m for which mN is a perfect square.
Show worked answer
Answer: 3
- A square has even prime exponents.
- Supply one additional copy of each prime whose exponent is odd: m=3.
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