Let N = (2^i)(3^j)(5^k) be such a number. Then there must exist numbers a, b, c so that:
- 2N = a2
- 3N = b3
- 5N = c5
The exponent i must be odd from (1), a multiple of 3 from (2) and a multiple of 5 from (3). The smallest exponent which meets these requirements is 15.
The exponent j must be even from (1), one less than a multiple of 3 from (2), and a multiple of 5 from (3). The smallest exponent which meets these requirements is 20.
The exponent k must be even from (1), a multiple of 3 from (2), and one less than a multiple of 5 from (3). The smallest exponent which meets these requirements is 24.
Therefore, the smallest such integer is N = (2^15)(3^20)(5^24) = 6,810,125,783,203,125,000,000,000,000,000.
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