8 (a) Two numbers, P and Q, are written as products of their prime factors - OCR - GCSE Maths - Question 8 - 2018 - Paper 4

First, we need to calculate the value of C:

$C = 2^3 × 3^1 × 5^2$

Now, we find P + C by adding their prime factorization:

P = $2^5 × 3^2 × 5^3 × 11^1$

C = $2^3 × 3^1 × 5^2$

To add them, we rewrite both in terms of the same base:

$P + C = 2^5 × 3^2 × 5^3 + 2^3 × 3^1 × 5^2$

Next, we factor out the common terms:

$P + C = 2^3 × 3^1 × 5^2 (2^2 × 3^1 × 5^1 + 1)$

Calculating in the brackets:

$2^2 × 3^1 × 5^1 = 4 × 3 × 5 = 60$

So, we have:

$P + C = 2^3 × 3^1 × 5^2 × 61$

To express 450 as a product of its prime factors, we can start by dividing 450 by its smallest prime factor:

$450 = 2 × 225$ $225 = 3 × 75$ $75 = 3 × 25$ $25 = 5 × 5$

Thus, combining these, we arrive at:

$450 = 2^1 × 3^2 × 5^2$

First, we need to find the prime factorization of both numbers:

For 270: $270 = 2^1 × 3^3 × 5^1$

For 450: $450 = 2^1 × 3^2 × 5^2$

The HCF is determined by taking the lowest powers of each common prime factor:

• For 2, the lowest power is $2^1$.
• For 3, the lowest power is $3^2$.
• For 5, the lowest power is $5^1$.

Thus, the HCF is:

$HCF = 2^1 × 3^2 × 5^1 = 90$