The prime factorisation of 450 is $2 \times 3^2 \times 5^2$. Express 648 as a product of its prime factors. Answer 648 = ... Using your answer to part (a), determine whether 583200 is a perfect square. ... Find the smallest integer $n$ such that $450n$ is a multiple of 648. Answer n = ...

1. Break 648 down into prime factors.

$648=2\times324=2\times2\times162=2\times2\times2\times81$
$81=3\times3\times3\times3$

2. Write the prime factorisation in index notation.

$648=2\times2\times2\times3\times3\times3\times3=2^3\times3^4$

1. Express 583200 using the given factorisation of 450 and the answer from part (a).

Since $583200=450\times1296$ and $1296=2\times648$,
$583200=450\times2\times648.$
Using $450=2\times3^2\times5^2$ and $648=2^3\times3^4$,
$583200=(2\times3^2\times5^2)\times2\times(2^3\times3^4).$

2. Combine the prime powers and decide whether the number is a perfect square.

$583200=2^{1+1+3}\times3^{2+4}\times5^2$
$=2^5\times3^6\times5^2.$
Since the power of $2$ is odd, not all indices are even.

1. Compare the prime factors of $450$ and $648$ to find the missing factors needed in $n$.

$450=2\times3^2\times5^2$ and, from part (a), $648=2^3\times3^4$.

For $450n$ to be a multiple of $648$, it must contain at least $2^3\times3^4$.

Since $450$ already contains $2\times3^2$, $n$ must contain
$2^{3-1}\times3^{4-2}=2^2\times3^2=36.$