Two integers S and T are written as products of their prime factors where p is a prime number. S = $3^2 \times 5 \times p$ T = $2^3 \times 3 \times p^2$ Lcm .

1. Identify the highest power of each prime factor needed for the LCM

$S=3^2\times 5\times p$
$T=2^3\times 3\times p^2$

For the smallest integer divisible by both $S$ and $T$, take the highest powers:

• $2^3$ from $T$
• $3^2$ from $S$
• $5$ from $S$
• $p^2$ from $T$

2. Write the LCM as a product of prime factors in terms of $p$

Hence,

$\operatorname{LCM}(S,T)=2^3\times 3^2\times 5\times p^2$

1. Make the powers in the prime factorisation of $S$ even

$S=3^2\times 5\times p$
For $S$ to be a perfect square, each prime factor must have an even power.

Since $3^2$ already has an even power, the factor $5$ must be paired with another $5$.

Hence $p=5.$

1. Write $525$ and the required HCF as products of prime factors

$525=3\times 5^2\times 7$
$105=3\times 5\times 7$

2. Compare the common prime factors of $S$ and $525$

Given
$S=3^2\times 5\times p.$
The common factors already include
$3\times 5.$
For the HCF to be
$105=3\times 5\times 7,$
$S$ must also contain the factor $7$.

Hence
$p=7.$