What is the smallest positive number that you can multiply by 2016 to make a perfect square?
2016 = 2^5 * 3^2 * 7
A perfect square has all powers even numbers, so you need to multiply by 2*7 = 14
That will give you 2^6 * 3^2 * 7^2 = (2^3 * 3 * 7)^2
To find the smallest positive number that can be multiplied by 2016 to make a perfect square, we need to determine the prime factorization of 2016.
First, we divide 2016 by its smallest prime factor, which is 2:
2016 ÷ 2 = 1008
Next, we divide 1008 by its smallest prime factor, which is 2:
1008 ÷ 2 = 504
Dividing 504 by 2 again:
504 ÷ 2 = 252
And dividing 252 by 2 again:
252 ÷ 2 = 126
Now, we divide 126 by its smallest prime factor, which is 3:
126 ÷ 3 = 42
Finally, we divide 42 by its smallest prime factor, which is 2:
42 ÷ 2 = 21
Now, we can write the prime factorization of 2016 as:
2016 = 2 × 2 × 2 × 2 × 3 × 3 × 7
To make a perfect square, we need each prime factor to appear an even number of times. Looking at the prime factorization, we notice that we have one extra 2 in comparison to the other prime factors. To make it a perfect square, we can multiply 2016 by one more 2:
2016 × 2 = 4032
So, the smallest positive number that can be multiplied by 2016 to make a perfect square is 4032.
To find the smallest positive number that you can multiply by 2016 to make a perfect square, we need to factorize 2016 and look for the largest possible exponent for each prime factor.
First, let's find the prime factorization of 2016:
2016 = 2 * 2 * 2 * 2 * 3 * 3 * 7
To make a perfect square, we need each prime factor to have an even exponent.
From the prime factorization, we have:
2^4 * 3^2 * 7^1
To make each exponent even, we need to raise 2 to the power of 2, 3 to the power of 2, and 7 to the power of 0 (since 7 has an odd exponent).
So, the smallest positive number we can multiply by 2016 to make a perfect square is:
2^2 * 3^2 * 7^0 = 4 * 9 * 1 = 36