Finding the Least Number for a Perfect Square

To find the least number by which 294 must be multiplied to make it a perfect square, we first find the prime factorization of 294.

Prime Factorization of 294

Step 1: Divide 294 by the smallest prime number, 2.

$294 \div 2 = 147$

Step 2: Divide 147 by the next smallest prime number, 3.

$147 \div 3 = 49$

Step 3: Divide 49 by the prime number 7.

$49 \div 7 = 7$

Step 4: Divide 7 by 7.

$7 \div 7 = 1$

The prime factorization of 294 is $2 \times 3 \times 7 \times 7$, which can be written as $2^1 \times 3^1 \times 7^2$.

Identifying the Multiplier

For a number to be a perfect square, all the exponents in its prime factorization must be even.

• The prime factor 2 has an exponent of 1 (odd).
• The prime factor 3 has an exponent of 1 (odd).
• The prime factor 7 has an exponent of 2 (even).

To make the exponents of 2 and 3 even, we need to multiply by one more 2 and one more 3.

The least number required is the product of the prime factors with odd exponents, raised to the power of 1.

Required multiplier $= 2 \times 3 = 6$.

Verification

Multiplying 294 by 6:

$294 \times 6 = (2^1 \times 3^1 \times 7^2) \times (2^1 \times 3^1)$

$= 2^{(1+1)} \times 3^{(1+1)} \times 7^2$

$= 2^2 \times 3^2 \times 7^2$

$= (2 \times 3 \times 7)^2$

$= 42^2$

Since $42^2$ is a perfect square, the least number by which 294 must be multiplied is 6.