8 (a) Two numbers, P and Q, are written as products of their prime factors. P = 2^5 × 3^2 × 5^3 × 11 Q = 2^4 × 3^5 × 7 (i) Find the lowest common multiple (LCM) of P and Q. (ii) The number C is written as the product of its prime factors. C = 2^3 × 3 × 5^2 Work out P + C, leaving your answer as a product of powers of prime numbers. (b) (i) Write 450 as a product of its prime factors. (ii) Find the highest common factor (HCF) of 270 and 450.

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

Question 8

8 (a) Two numbers, P and Q, are written as products of their prime factors. P = 2^5 × 3^2 × 5^3 × 11 Q = 2^4 × 3^5 × 7 (i) Find the lowest common multiple (LCM) o... show full transcript

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

Step 1

Find the lowest common multiple (LCM) of P and Q.

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Answer

To determine the LCM of P and Q, we need to identify the highest powers of each prime factor present in both numbers:

For the prime number 2, the highest power is $2^5$ (from P).
For the prime number 3, the highest power is $3^5$ (from Q).
For the prime number 5, the highest power is $5^3$ (from P).
For the prime number 7, the highest power is $7^1$ (from Q).
For the prime number 11, the highest power is $11^1$ (from P).

Thus, the LCM is given by:

$LCM = 2^5 × 3^5 × 5^3 × 7^1 × 11^1$

Step 2

Work out P + C, leaving your answer as a product of powers of prime numbers.

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Answer

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:

$P + C = 2^3 × 3^1 × 5^2 × 61$

Step 3

Write 450 as a product of its prime factors.

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Answer

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:

$450 = 2^1 × 3^2 × 5^2$

Step 4

Find the highest common factor (HCF) of 270 and 450.

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Answer

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:

$HCF = 2^1 × 3^2 × 5^1 = 90$

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

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

Find the lowest common multiple (LCM) of P and Q.

Work out P + C, leaving your answer as a product of powers of prime numbers.

Write 450 as a product of its prime factors.

Find the highest common factor (HCF) of 270 and 450.

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