+816 How many positive perfect square integers are factors of the product \(\left(2^{10}\right)\left(3^{12}\right)\left(5^{15}\right)\)?
What is the smallest positive integer n such that, out of the n unit fractions \(\frac{1}{k}\)where \(1 \le k \le n\), exactly half of the fractions give a terminating decimal?
I think this is the number of perfect squares:
2^5 x 3^6 x 5^7.5 =4,075,233,888 Perfect squares??
(2^10)(3^12)(5^15) =16,607,531,250,000,000,000
Number of perfect squares that are FACTORS of the above product are calculated as follows:
The exponent of each term / 2 + 1, then multiplied together as follows:
2^10 =10 / 2 + 1 = 6, 3^12 =12/2 + 1 =7, 5^15 =15/2 + 1 =8
The total number of perfect squares that are FACTORS of the above product:
=6 x 7 x 8 =336 perfect squares.
