What is the smallest positive integer N such that 2N is a perfect square 3N is a perfect cube and 5N is a perfect fifth?

Answer 1

For a number to be a perfect square, its prime factors must be separable into two identical sets. For a number to be a perfect cube, its prime factors must be separable into three identical sets. For a number to be a perfect 'fifth', its prime factors must be separable into five identical sets. ---- If N x 2 is separable into two identical sets of prime factors, then: * N must have an odd number of 2's in its prime factorisation. * All other prime factors of N must occur an even number of times. If N x 3 is separable into three identical sets of prime factors, then: * The prime factorisation of N must contain a number of 3's that is one less than a multiple of three. * All other prime factors of N must occur a number of times that is already a multiple of three. If N x 5 is separable into five identical sets of prime factors, then: * The prime factorisation of N must contain a number of 5's that is one less than a multiple of five. * All other prime factors of N must occur a number of times that is already a multiple of five. ---- From the above, we know that N's prime factorisation must contain 2's, 3's and 5's. How many times does 2 appear in the prime factorisation of N? - It must be an odd number. - It must be a multiple of three. - It must be a multiple of five. The smallest valid frequency is 15. How many times does 3 appear in the prime factorisation of N? - It must be an even number. - It must be one less than a multiple of three. - It must be a multiple of five. The smallest valid frequency is 20. How many times does 5 appear in the prime factorisation of N? - It must be an even number. - It must be a multiple of three. - It must be one less than a multiple of five. The smallest valid frequency is 24. ---- Therefore N = 215 x 320 x 524 = 6810125783203125000000000000000. That seems to be the smallest possible value.