What is the smallest positive integer by which 180 should be multiplied to obtain a perfect square?
A perfect square will have an even number of repetitions of each prime factor.
180 = 2*2*3*3*5
e have 2^2 and 3^2, but we need another 5. So, 180*5 = 900 = 30^2 = (2*3*5)^2
To find the smallest positive integer by which 180 should be multiplied to obtain a perfect square, we need to analyze the prime factorization of 180.
The prime factorization of 180 can be found by dividing it by prime numbers starting from 2:
180 ÷ 2 = 90
90 ÷ 2 = 45
45 ÷ 3 = 15
15 ÷ 3 = 5
So, 180 can be expressed as 2 × 2 × 3 × 3 × 5.
To obtain a perfect square, we need each prime factor to appear an even number of times.
From the prime factorization of 180, we have two 2's and two 3's. To make them appear an even number of times, we need to multiply 180 by another 2 and another 3.
Therefore, the smallest positive integer by which 180 should be multiplied to obtain a perfect square is 2 × 3 = 6.