Question

What is the lowest common multiple of 36 and 56 that is also a square number?

Answer 1

The lowest common multiple of 36 and 56 that is also a square number is 7056, obtained by multiplying the least common multiple (LCM), 504, by additional factors to obtain all even-power prime factors.

Step-by-step explanation:

To find the lowest common multiple of 36 and 56 that is also a square number, we first need to find the least common multiple (LCM) of the two numbers. The prime factorization of 36 is 2² × 3² and of 56 is 2³ × 7. The LCM is obtained by multiplying the highest power of all the prime factors that appear in either number, which gives us 2³ × 3² × 7 = 504. However, 504 is not a square number. To get the lowest common multiple that is also a square, we need to multiply 504 by the least amount necessary to make each of the exponents an even number. We already have 3² which is the square of 3, but we need two 2's and two 7's to make squares, which means we need to multiply by 2± (to make 2´) and 7± (to make 7²). Thus, the lowest common multiple of 36 and 56 that is a square number is 2´ × 3² × 7² = 7056.

Answer 2

The lowest common multiple of 36 and 56 that is also a square number is 28224. This is found by making sure the prime factorization of the LCM has even exponents, resulting in a square number: 2´ × 3² × 7².

Step-by-step explanation:

To find the lowest common multiple of 36 and 56 that is also a square number, we need to first find the prime factorization of both numbers. The prime factorization of 36 is 2² × 3², and for 56, it is 2³ × 7. The least common multiple (LCM) of two numbers is found by multiplying each prime number the greatest number of times it appears in either factorization.

For 36 and 56, the LCM is 2³ × 3² × 7, which is 1008. Since we are looking for the LCM that is also a square number, we need to find the smallest square number that is a multiple of 1008. The prime factors need to be raised to an even power for the result to be a square. 1008 = 2³ × 3² × 7 does not have all even exponents, so we need to increase the powers of 2 and 7 to make them even. By doing this, we obtain 2´ × 3² × 7², which is 28224, and that is the smallest square number that is a multiple of both 36 and 56.