Number System (Classes VI-VIII) • Topic 2 of 6

Playing with Numbers (Factors, Multiples, HCF, LCM)

Factors, multiples, prime and composite numbers, prime factorisation, and the HCF and LCM live here, and this is the richest source of CTET misconception items in the whole chapter. The two errors examiners adore: a child who computes the HCF by multiplying ALL the common factors (so for 12 and 18 they multiply 2 x 3 x 6 instead of taking the right product of lowest prime powers, 2 x 3 = 6), and a child who confuses HCF with LCM, giving the larger answer when the smaller is wanted. A teacher must show that HCF takes the LOWEST powers of the shared primes while LCM takes the HIGHEST powers of every prime present. Other staples: 1 is neither prime nor composite, 2 is the only even prime, and the difference between a factor (divides into) and a multiple (built up from). CTET frames these as word problems, where 'largest size that fits both' signals HCF and 'they meet again / ring together' signals LCM.

✅ Solved examples

1. Find the HCF and LCM of 12 and 18 using prime factorisation.
12 = 2^2 x 3 and 18 = 2 x 3^2. HCF takes lowest powers: 2^1 x 3^1 = 6. LCM takes highest powers: 2^2 x 3^2 = 36. Check: 6 x 36 = 216 = 12 x 18.
2. A child finds the HCF of 12 and 18 as 36. What error has the child made?
The child has found the LCM (highest common multiple thinking) and labelled it HCF. HCF is the highest common FACTOR, which must be 6 and is at most the smaller number, 12. The child has swapped the two ideas.
3. What is the largest number that divides both 24 and 36 exactly?
This is the HCF. 24 = 2^3 x 3, 36 = 2^2 x 3^2, so HCF = 2^2 x 3 = 12.
4. Two bells ring at intervals of 9 and 12 minutes. After how many minutes do they ring together?
This is an LCM problem. 9 = 3^2, 12 = 2^2 x 3, so LCM = 2^2 x 3^2 = 36 minutes.

✏️ Practice — try these, take hints as needed

1. Find the HCF of 36 and 48.
Prime factorise both.
Take the lowest power of each common prime.
12 (36 = 2^2 x 3^2, 48 = 2^4 x 3, HCF = 2^2 x 3 = 12)
2. A teacher asks for the largest square tile that can pave a floor 18 m by 24 m with no cutting. Which concept and what is the answer?
Largest size that fits both means HCF.
Factorise 18 and 24.
HCF; 6 metres (18 = 2 x 3^2, 24 = 2^3 x 3, HCF = 2 x 3 = 6)
3. Is 1 a prime number? What should a teacher tell the class?
A prime has exactly two distinct factors.
Count the factors of 1.
No. 1 has only one factor (itself), so it is neither prime nor composite.
4. Find the LCM of 8 and 10.
8 = 2^3, 10 = 2 x 5.
Take the highest power of every prime.
40

📝 Topic test — 8 questions

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