Use prime factors to explain why 56×126 is a perfect square. Answer ... ... Write down the greatest integer that will divide both 56 and 126 exactly. Answer ... The number 126k is a perfect cube. Find the smallest positive integer value of k. Answer ...

1. Write 56 and 126 as products of prime factors.

$56 = 2^3 \times 7$

$126 = 2 \times 3^2 \times 7$

2. Multiply the prime factorisations and combine like prime factors.

$56 \times 126 = (2^3 \times 7)(2 \times 3^2 \times 7)$

$= 2^4 \times 3^2 \times 7^2$

3. Explain why the product is a perfect square.

In $2^4 \times 3^2 \times 7^2$, all the powers are even.

So $56 \times 126 = (2^2 \times 3 \times 7)^2 = 84^2$, which is a perfect square.

1. Find the greatest integer that divides both numbers exactly.

From part (a),
$56=2^3\times 7$ and $126=2\times 3^2\times 7$.

The common prime factors are $2$ and $7$, so
$\text{HCF}=2\times 7=14.$

1. Write 126 as a product of prime factors.

$126=2\times 3^2\times 7$

2. Make all prime powers have exponents that are multiples of 3.

For $126k$ to be a perfect cube, each prime factor must have an exponent that is a multiple of $3$.

$126=2^1\times 3^2\times 7^1$

So multiply by
$2^2\times 3^1\times 7^2$

Hence
$k=2^2\times 3\times 7^2=4\times 3\times 49=588.$