Written as the product of its prime factors, 360=2^3 times 3^2 times 5. So find the smallest positive integer k such that 360k is a cube number.
you need all the powers to be 3, so
360 * 3*5^2 = 2^3 3^3 5^3
To find the smallest positive integer k such that 360k is a cube number, we need to determine the prime factors of 360 and then find the smallest value of k that will make the product of those prime factors a perfect cube.
The prime factorization of 360 is: 2^3 * 3^2 * 5.
For a number to be a perfect cube, each of its prime factors must have an exponent that is a multiple of 3. In other words, the exponents of the prime factors in the prime factorization must be divisible by 3.
Let's examine each prime factor and its exponent:
- The exponent of 2 is 3. Since 3 is already divisible by 3, we don't need to make any changes to the exponent of 2.
- The exponent of 3 is 2. Since 2 is not divisible by 3, we need to increase the exponent of 3 to the next multiple of 3, which is 3 itself. So, we need to multiply 360 by an extra 3 to make the exponent of 3 divisible by 3.
- The exponent of 5 is 1. Since 1 is not divisible by 3, we need to increase the exponent of 5 to the next multiple of 3, which is 3. So, we need to multiply 360 by an extra 5^2 (which is 25) to make the exponent of 5 divisible by 3.
By multiplying 360 by 3 (to account for the exponent of 3) and by 25 (to account for the exponent of 5), we get:
360 * 3 * 25 = 27,000.
Therefore, the smallest positive integer k such that 360k is a cube number is 27,000.