Rational Numbers | FIO

Question 6

Find the smallest number by which 9408 must be multiplied so that the product is a perfect square. Find the square root of the product.

Solution

Understand the Question

A number is a perfect square if all prime factors in its prime factorisation occur in complete pairs (even powers).
To find the smallest number to multiply $9408$ by, we find its prime factorisation and identify any unpaired factor.
Multiplying $9408$ by the unpaired factor makes all prime factors paired, giving a perfect square.
We then take one factor from each pair to find the square root of the new product.

Step 1 · Prime Factorisation of 9408

Diagram 1

\begin{aligned} 9408 \div 2 &= 4704 \\ 4704 \div 2 &= 2352 \\ 2352 \div 2 &= 1176 \\ 1176 \div 2 &= 588 \\ 588 \div 2 &= 294 \\ 294 \div 2 &= 147 \\ 147 \div 3 &= 49 \\ 49 \div 7 &= 7 \\ 7 \div 7 &= 1 \end{aligned}

Prime factorisation of $9408$ : $9408 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 7 \times 7$

Grouping into pairs: $9408 = (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times 3 \times (7 \times 7)$

The prime factor $3$ is unpaired. Therefore, $9408$ must be multiplied by $3$ to make it a perfect square.

Step 2 · Calculate the Product and its Square Root

Multiply $9408$ by $3$ :

\begin{aligned} \text{Product} &= 9408 \times 3 \\[0.6em] &= 28224 \end{aligned}

Prime factors of $28224$ : $28224 = (2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (3 \times 3) \times (7 \times 7)$

Taking one factor from each pair:

\begin{aligned} \sqrt{28224} &= \sqrt{(2 \times 2) \times (2 \times 2) \times (2 \times 2) \times (3 \times 3) \times (7 \times 7)} \\[0.6em] &= 2 \times 2 \times 2 \times 3 \times 7 \\[0.6em] &= 8 \times 21 \\[0.6em] &= 168 \end{aligned}

Answer

Smallest multiplying factor is $3$ , and $\sqrt{28224} = 168$

Common Mistakes

Arithmetic Error in Division: Making mistakes during repeated division, particularly when dividing $147$ by $3$ or $49$ by $7$ .
Pairing Error: Forgetting to include the newly multiplied factor $3$ when pairing the prime factors for the square root calculation.
Taking Square Root Directly: Attempting to find the square root of $9408$ instead of the resulting product ( $28224$ ).

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