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For perfect squares, the powers of each prime number must be a multiple of 2 (so that the prime numbers can be split equally into two brackets).
For perfect cubes, the powers of each prime number must be a multiple of 3 (so that the prime numbers can be split equally into three brackets).
For numbers which are both perfect squares and perfect cubes, the powers of each prime number must be a multiple of 2 AND 3 i.e. a multiple of 6 (so that the prime numbers can be split equally into six different brackets).
360 = 2^3 x 3^2 x 5.
We need each power of the prime number to be a multiple of 6, minimally 6.
To do so, we need three more '2's, four more '3's and five more '5's. This must come from the k.
So k is the product of the three '2's, four '3's and five '5's, totalling 2025000.
For perfect cubes, the powers of each prime number must be a multiple of 3 (so that the prime numbers can be split equally into three brackets).
For numbers which are both perfect squares and perfect cubes, the powers of each prime number must be a multiple of 2 AND 3 i.e. a multiple of 6 (so that the prime numbers can be split equally into six different brackets).
360 = 2^3 x 3^2 x 5.
We need each power of the prime number to be a multiple of 6, minimally 6.
To do so, we need three more '2's, four more '3's and five more '5's. This must come from the k.
So k is the product of the three '2's, four '3's and five '5's, totalling 2025000.
Date Posted:
5 years ago