NCERT Solutions for Class 7th Maths Chapter 3 In-text Questions — Efficient Procedures for HCF and LCM

Book page 60–61 Updated on2026-09-19

Q1.

See the procedure on the right. Can you explain how it has been carried out? [2 | 84, 180 · 2 | 42, 90 · 3 | 21, 45 · 7, 15]

Answer

Both numbers are divided by a common prime at each step, and the two quotients are written in the next row.

2 | 84, 180   → 84 ÷ 2 = 42 and 180 ÷ 2 = 90
2 | 42, 90    → 42 ÷ 2 = 21 and 90 ÷ 2 = 45
3 | 21, 45    → 21 ÷ 3 = 7 and 45 ÷ 3 = 15
     7, 15      → 7 and 15 have no common prime factor, so we stop

This is the division method of prime factorisation, run on two numbers at once.

Why it happens: at every step the divisor must divide both numbers, so it is a common factor. We stop when the two numbers left share nothing — 7 is prime and does not divide 15. Reading the work back gives 84 = 2 × 2 × 3 × 7 and 180 = 2 × 2 × 3 × 15.

Q2.

How do we use this to find the HCF of 84 and 180? Explore. [Hint: Observe that 84 = 2 × 2 × 3 × 7, and 180 = 2 × 2 × 3 × 15 similar to prime factorisation]

Answer

Multiply the numbers in the left column.

Left column: 2, 2, 3
HCF(84, 180) = 2 × 2 × 3 = 12
Check: 84 ÷ 12 = 7 ✓ and 180 ÷ 12 = 15 ✓

Why it happens: the hint spells it out — the ladder rewrites the numbers as 84 = (2 × 2 × 3) × 7 and 180 = (2 × 2 × 3) × 15. The bracket, which is the left column, is a common factor of both. And it is the highest one, because we only stopped when the leftovers 7 and 15 had nothing more in common. Anything still shared would have given one more row.

Tip: this is much quicker than factorising each number separately and comparing — one ladder does both numbers at once.

Q3.

Find the HCF in the following cases. [2 | 300, 150 · 5 | 150, 75 · 5 | 30, 15 · 3 | 6, 3 · 2, 1 — HCF = 2 × 5 × 5 × 3] and [2 | 630, 770 · 5 | 315, 385 · 7 | 63, 77 · 9, 11 — HCF = 2 × 5 × 7]

Answer

Multiply out the left column of each ladder.

300 and 150
2 | 300, 150 → 150, 75
5 | 150, 75 → 30, 15
5 | 30, 15 → 6, 3
3 | 6, 3 → 2, 1
2 and 1 have no common factor — stop
HCF = 2 × 5 × 5 × 3 = 150

630 and 770
2 | 630, 770 → 315, 385
5 | 315, 385 → 63, 77
7 | 63, 77 → 9, 11
9 and 11 have no common factor — stop
HCF = 2 × 5 × 7 = 70

Checks: 300 ÷ 150 = 2 and 150 ÷ 150 = 1 ✓; 630 ÷ 70 = 9 and 770 ÷ 70 = 11 ✓

Tip: for 300 and 150 the HCF is 150, one of the two numbers — as expected, since 150 is a factor of 300.

Q4.

This procedure not only gives the HCF but can also be used to find the LCM! Can you see how?

Answer

Multiply the left column together with the last row — the whole L-shape.

300 and 150: left column 2, 5, 5, 3 and last row 2, 1
LCM = 2 × 5 × 5 × 3 × 2 × 1 = 300

630 and 770: left column 2, 5, 7 and last row 9, 11
LCM = 2 × 5 × 7 × 9 × 11 = 70 × 99 = 6930

Note that for 630 and 770 the book carries the ladder one step further, dividing 9 by 3 to get 3 — which is why it writes LCM = 2 × 5 × 7 × 3 × 3 × 11. That is the same number: 3 × 3 = 9.

Why it happens: the ladder splits each number into (common part) × (leftover). Here 630 = 70 × 9 and 770 = 70 × 11. A common multiple must contain the common part 70, the leftover 9 and the leftover 11 — and once the leftovers share nothing, that is all it needs.

Q5.

Why are these the LCMs? [Hint: Will the product of the factors marked as the LCM of 300 and 150 contain the prime factorisations of both 300 and 150? Is this the smallest such number?]

Answer

Because the L-shaped product contains both numbers, and nothing in it can be dropped.

Does it contain both?
2 × 5 × 5 × 3 × 2 × 1 = 300
Reading 300 = (2 × 5 × 5 × 3) × 2 — the left column with the 2 from the last row ✓
Reading 150 = (2 × 5 × 5 × 3) × 1 — the left column with the 1 from the last row ✓
So 300 is a common multiple of 300 and 150

Is it the smallest?
Left column = HCF = 150, leftovers 2 and 1
Any common multiple must hold the HCF 150, the leftover 2 and the leftover 1
The leftovers share nothing, so none of them can be spared
Smallest = 150 × 2 × 1 = 300

Same reasoning for 630 and 770: HCF 70, leftovers 9 and 11, which are co-prime. LCM = 70 × 9 × 11 = 6930.

Why it happens: write the two numbers as h × p and h × q, where h is the HCF and p, q are the leftovers. The number h × p × q is a multiple of both (it is h × p times q, and h × q times p). It cannot be made smaller: dropping anything from p would stop it being a multiple of h × p, and dropping anything from q would stop it being a multiple of h × q. So h × p × q is the LCM. Notice too that HCF × LCM = h × (h × p × q) = (h × p) × (h × q) = the product of the two numbers.

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