NCERT Solutions for Class 7th Maths Chapter 3 In-text Questions — Property Involving both the HCF and the LCM

Book page 62–63 Updated on2026-09-19

Q1.

Which is greater — the LCM of two numbers or their product?

Answer

The product is greater, or the two are equal. The LCM is never bigger than the product.

Numbers	Product	LCM	Which is greater?
6, 8	48	24	product
12, 18	216	36	product
7, 11	77	77	equal
4, 9	36	36	equal
15, 25	375	75	product

They are equal exactly when the two numbers are co-prime.

Q2.

You could analyse the above statement using examples. Then try to reason or prove, why the LCM is never greater than the product of the numbers. [Hint: Is the product also a common multiple of the two numbers?]

Answer

Because the product is itself a common multiple, and the LCM is the smallest common multiple.

Take the two numbers a and b
a × b = a × b, so a × b is a multiple of a ✓
a × b = b × a, so a × b is a multiple of b ✓
So a × b is a common multiple of a and b
The LCM is the smallest common multiple
Therefore LCM ≤ a × b

Why it happens: the LCM has to be less than or equal to every common multiple, since it is the least of them. The product is one of the common multiples on that list. So the LCM cannot overtake it. The two are equal only when the product is itself the smallest — that is, when a and b share no prime factor and nothing can be trimmed away.

Tip: in prime terms, the product counts every shared prime twice while the LCM counts it once. The extra copies are exactly the HCF — which is why LCM = (a × b) ÷ HCF.

Q3.

Consider the numbers 105 and 95. Find their LCM. [Is the LCM a factor of the product? If yes, what should it be multiplied with to get the product?]

Answer

LCM(105, 95) = 1995, and it is a factor of the product — multiply it by 5 to get the product.

105 = 3 × 5 × 7
95 = 5 × 19
LCM = 3 × 5 × 7 × 19 = 1995

Product in factorised form: 105 × 95 = 3 × 5 × 5 × 7 × 19 = 9975
Comparing: 3 × 5 × 5 × 7 × 19 = (3 × 5 × 7 × 19) × 5
So 105 × 95 = LCM × 5
Check: 1995 × 5 = 9975 ✓

And 5 is exactly HCF(105, 95).

Why it happens: both numbers carry a 5. When you multiply them, that 5 is counted twice; the LCM keeps only one copy. The one spare 5 is what is left over — and one spare copy of each shared prime is precisely the HCF.

Q4.

Explore whether the LCM is a factor of the product in the following cases. If yes, identify the number that the LCM should be multiplied by to get the product. Do you see any pattern? Use these numbers: (a) 45, 105 (b) 275, 352 (c) 222, 370

Answer

In every case the LCM is a factor of the product, and the multiplier is the HCF.

	Prime factorisations	LCM	Product	Multiplier
(a)	45 = 3 × 3 × 5 105 = 3 × 5 × 7	3 × 3 × 5 × 7 = 315	4725	15 = HCF
(b)	275 = 5 × 5 × 11 352 = 2 × 2 × 2 × 2 × 2 × 11	2⁵ × 5 × 5 × 11 = 8800	96800	11 = HCF
(c)	222 = 2 × 3 × 37 370 = 2 × 5 × 37	2 × 3 × 5 × 37 = 1110	82140	74 = HCF

(a) 4725 ÷ 315 = 15, and HCF(45, 105) = 3 × 5 = 15 ✓
(b) 96800 ÷ 8800 = 11, and HCF(275, 352) = 11 ✓
(c) 82140 ÷ 1110 = 74, and HCF(222, 370) = 2 × 37 = 74 ✓

The pattern: product ÷ LCM = HCF, that is, HCF × LCM = product.

Q5.

Do you see that, in each case, the number by which the LCM is multiplied to get the product is actually the HCF?

Answer

Yes. In every case the multiplier turned out to be the HCF.

Numbers	HCF	LCM	HCF × LCM	Product
105, 95	5	1995	9975	9975 ✓
45, 105	15	315	4725	4725 ✓
275, 352	11	8800	96800	96800 ✓
222, 370	74	1110	82140	82140 ✓

HCF × LCM = Product of the two numbers

Tip: this gives a quick way to find one from the other. If you know HCF(84, 132) = 12, then LCM = (84 × 132) ÷ 12 = 11088 ÷ 12 = 924 — no factorising needed.

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