23X5
The prime factorization of 40 is 2x2x2x5 or 23x5.
The prime factorization of 32 is 2x2x2x2x2 or 25. The prime factorization of 40 is 2x2x2x5 or 23x5. For a number to be a multiple of a given number the multiple must have the entire prime factorization of that number in is. So a multiple of 32 must contain 2x2x2x2x2 or 25 . Likewise a multiple of 40 must contain 2x2x2x5 or 23x5. The least number that this could possible happen with must have 5 factors of 2 and one factor of 5 (the highest exponent on each individual prime factor). Thus the LCM is 2x2x2x2x2x5 or 25x5 Which is 160.
Boy, I wish this website could notate exponents properly. If that's 24 x 3 x 5 and 23 x 52, the GCF is indeed 23 x 5. Good job. In other words, the GCF of 240 and 200 is 40.
The integer that is a perfect square, perfect cube, and perfect fifth power is 64. This is because 64 is equal to 8^2, 4^3, and 2^6, making it a perfect square, cube, and fifth power. It is the only integer between 2 and 200000000000000 that satisfies all three conditions.