To find the number that should multiply 117 to make it a perfect square, we first need to understand what a perfect square is. A perfect square is an integer that is the square of another integer. For example, 9 is a perfect square because it is \(3^2\). When we express a perfect square in terms of its prime factors, all the exponents of its prime factors must be even.

Let's follow these steps to find the smallest number to multiply by 117:

Prime Factorization of 117

The first crucial step is to find the prime factors of the given number, 117. We break down 117 into its prime components using division:

• Divide 117 by the smallest prime number, 3.
• \(117 \div 3 = 39\)
• Now, divide 39 by 3 again.
• \(39 \div 3 = 13\)
• 13 is a prime number, so we stop the factorization process here.

Therefore, the prime factorization of 117 is \(3 \times 3 \times 13\). We can write this in exponential form as:

\(117 = 3^2 \times 13^1\)

Perfect Square Condition

For any number to be considered a perfect square, the exponent of each of its prime factors must be an even number. Let's look at the exponents in the prime factorization of 117 to identify any odd exponents:

• The prime factor 3 has an exponent of 2, which is an even number. This part (\(3^2\)) is already in a perfect square form.
• The prime factor 13 has an exponent of 1, which is an odd number. To make this exponent even, we need to multiply \(13^1\) by another 13. This would result in \(13^1 \times 13^1 = 13^{(1+1)} = 13^2\).

This analysis shows that 13 is the single missing factor needed to make 117 a perfect square.

Multiplying to Achieve a Perfect Square

To transform 117 into a perfect square, we need to multiply it by the identified missing factor, which is 13:

\(117 \times 13 = (3^2 \times 13^1) \times 13^1\)

Using the rules of exponents, we combine the terms with the same base:

\(= 3^2 \times 13^{(1+1)}\)

\(= 3^2 \times 13^2\)

Now, both prime factors (3 and 13) have even exponents (2). This confirms that the resulting number is a perfect square. We can also write this as:

\(3^2 \times 13^2 = (3 \times 13)^2 = 39^2\)

So, \(117 \times 13 = 1521\), and \(1521\) is indeed a perfect square because \(39 \times 39 = 1521\).

Thus, the smallest number by which 117 should be multiplied to make it a perfect square is 13.