\(84 = 2^2 \times 3 \times 7\) and \(90 = 2 \times 3^2 \times 5\) The Venn diagram shows some of the prime factors of 84 and 90. 84 90 2 7 2 3 ? 5 © EAGLE B...
Question 1 Report
\(84 = 2^2 \times 3 \times 7\) and \(90 = 2 \times 3^2 \times 5\)
The Venn diagram shows some of the prime factors of 84 and 90.
- Write down the value that should replace the question mark. (1)
- Use the Venn diagram to find the HCF of 84 and 90. (1)
- Use the Venn diagram to find the LCM of 84 and 90. (1)
Answer Details
This question tests using a Venn diagram of prime factors to find the highest common factor (HCF) and lowest common multiple (LCM) of two numbers.
(a) The circle for 90 must multiply to give \(90 = 2 \times 3^2 \times 5\). The diagram already shows an overlap of \(2\) and \(3\) shared with 84, together with a \(5\) in the 90-only region, so the missing value must supply the second factor of 3: \(2 \times 3 \times 3 \times 5 = 90\). The missing value is \(3\). [1]
(b) The HCF is the product of the numbers in the overlap of the two circles, since these are exactly the prime factors common to both 84 and 90:
\[\text{HCF} = 2 \times 3 = 6\][1]
(c) The LCM is the product of every number in the diagram, counting each region once:
\[\text{LCM} = 2 \times 7 \times 2 \times 3 \times 3 \times 5 = 1260\]This agrees with taking the highest power of each prime from \(84 = 2^2 \times 3 \times 7\) and \(90 = 2 \times 3^2 \times 5\): \(2^2 \times 3^2 \times 5 \times 7 = 1260\). [1]
The overlap of a prime-factor Venn diagram always gives the HCF, and multiplying every region together always gives the LCM; this is quicker and less error-prone than listing multiples of each number.
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