What is the smallest integer I can multiply 315 by to get a perfect square?
315 = 3^2 * 5 * 7
to be a perfect square, all the powers must be even. So, multiply by
5 * 7 = 35
to get 11025 = 105^2
To find the smallest integer you can multiply 315 by to get a perfect square, you need to consider the prime factorization of 315.
The prime factorization of 315 is 3 × 3 × 5 × 7.
A perfect square is a number that can be expressed as the product of an integer with itself. In order for a number to be a perfect square, its prime factors must all have even exponents.
So, to find the smallest integer you can multiply 315 by to get a perfect square, you need to determine the minimum number of additional factors of each prime number from the prime factorization to make all the exponents even.
In the case of 315, we have two 3's, one 5, and one 7.
Since two 3's already have an even exponent, we don't need to introduce any additional 3's.
Similarly, one 5 and one 7 already have odd exponents, so we need to introduce one additional factor of 5 and one additional factor of 7 to make their exponents even.
Therefore, the smallest integer you can multiply 315 by to get a perfect square is:
3 × 3 × 5 × 7 × 5 × 7 = 3^2 × 5^2 × 7^2 = 3,675.
So, 3,675 is the smallest integer you can multiply 315 by to get a perfect square.
To find the smallest integer that can be multiplied by 315 to get a perfect square, we can start by finding the prime factorization of 315.
The prime factorization of 315 is: 3 * 3 * 5 * 7.
To get a perfect square, all the exponents of the prime factors in the prime factorization should be even.
Since we have only one factor of 5 and one factor of 7, we need to multiply 315 by another factor of 5 and another factor of 7 to make the exponents of both 5 and 7 even.
Therefore, the smallest integer you can multiply 315 by to get a perfect square is: 3 * 3 * 5 * 5 * 7 * 7 = 6615.