Finding the Least Number to Create a Perfect Square

The question asks for the smallest positive whole number that we can multiply by 28 to get a perfect square. A perfect square is a number that can be obtained by squaring an integer (e.g., \(4=2^2\), \(9=3^2\), \(16=4^2\), etc.).

Understanding Perfect Squares through Prime Factorization

A key property of perfect squares is related to their prime factorization. When you find the prime factors of a perfect square, the exponent of each prime factor is always an even number.

For example:

• \(36 = 6^2 = (2 \times 3)^2 = 2^2 \times 3^2\). The exponents (2 and 2) are even.
• \(100 = 10^2 = (2 \times 5)^2 = 2^2 \times 5^2\). The exponents (2 and 2) are even.
• \(144 = 12^2 = (2^2 \times 3)^2 = (2^2)^2 \times 3^2 = 2^4 \times 3^2\). The exponents (4 and 2) are even.

Prime Factorization of 28

Let's find the prime factorization of 28:

\(\qquad 28 = 2 \times 14\)

\(\qquad 28 = 2 \times 2 \times 7\)

\(\qquad 28 = 2^2 \times 7^1\)

Looking at the prime factorization \(2^2 \times 7^1\), the prime factor 2 has an exponent of 2 (which is even), but the prime factor 7 has an exponent of 1 (which is odd).

Finding the Missing Factor

To make the number \(28 \times (\text{some number})\) a perfect square, all the exponents in its prime factorization must be even. In \(2^2 \times 7^1\), the exponent of 7 needs to become an even number. The smallest even number greater than 1 is 2.

To change the exponent of 7 from 1 to 2, we need to multiply \(7^1\) by \(7^1\). So, the missing factor needed is 7.

Let's check this:

\(\qquad 28 \times 7 = (2^2 \times 7^1) \times 7\)

\(\qquad = 2^2 \times (7^1 \times 7^1)\)

\(\qquad = 2^2 \times 7^{(1+1)}\)

\(\qquad = 2^2 \times 7^2\)

The prime factorization \(2^2 \times 7^2\) has both exponents (2 and 2) as even numbers. Therefore, \(2^2 \times 7^2\) is a perfect square.

\(\qquad 2^2 \times 7^2 = (2 \times 7)^2 = 14^2 = 196\)

196 is indeed a perfect square.

The least number we multiplied by 28 to get 196 was 7.

Checking the Options

Let's see what happens when we multiply 28 by the numbers given in the options:

The least number among the options which makes 28 a perfect square when multiplied is 7.

Step-by-Step Solution

1. Find the prime factorization of 28.
2. Identify the exponents of each prime factor.
3. Determine which prime factors have odd exponents.
4. To make the number a perfect square, multiply by the prime factors that have odd exponents, raised to the power needed to make the exponent even (usually 1, to turn \(x^1\) into \(x^2\)).
5. The product of these needed prime factors is the least number required.

For 28:

1. Prime factorization of 28 is \(2^2 \times 7^1\).
2. Exponents are 2 (for 2) and 1 (for 7).
3. The prime factor 7 has an odd exponent (1).
4. To make the exponent of 7 even (2), we need to multiply by \(7^1\).
5. The least number is 7.

Revision Table: Perfect Squares and Prime Factorization

Additional Information: Properties of Perfect Squares

Understanding perfect squares is fundamental in number theory. Here are a few more points:

• Perfect squares always end in 0, 1, 4, 5, 6, or 9 in base 10. They can never end in 2, 3, 7, or 8.
• The square root of a perfect square is an integer.
• The number of divisors of a perfect square is always odd.
• Perfect squares can be represented as the sum of consecutive odd numbers, starting from 1 (e.g., \(1=1^2\), \(1+3=4=2^2\), \(1+3+5=9=3^2\)).

Using prime factorization to determine if a number is a perfect square or to find the missing factor to make it a perfect square is a very efficient method.