Question

FIND two DIGIT Number WHOSE HCF IS 24 AND LCM IS 144

Answer 1

Solution :-

Given - L.C.M. is 144 and H.C.F. is 24

Let us assume that the two numbers be a and b respectively.

We know that,

L.C.M.*H.C.F. = Product of the two numbers

Then, 

L.C.M. = (a*b)/H.C.F.

⇒ 144 = ab/H.C.F.

⇒ ab = 144*24

⇒ ab = 3456 = 2⁷ × 3³

Since H.C.F. (ab) = 24 = 2³*3

And, 

Since, L.C.M. depends on ab and H.C.F. and we have ab and H.C.F. as 3456 and 24 respectively,

We can say that (a, b) = (24,144) ; (48, 72) ; (72, 48) ; (144, 24)

One of the two digit numbers can be 24, or, 48 or 72 whose L.C.M. is 144 and H.C.F. is 24 respectively.

Answer.

Answer 2

Given :

L.C.M= 144

H.C.F. = 24

Let the two numbers be p & q

L.C.M × H.C.F = Product of the two numbers

L.C.M = Product of the two numbers/ H.C.F

L.C.M. = (p×q)/H.C.F.


144 = pq /24

pq= 144× 24

pq= (2×2×2×2×3×3)× (2×2×2×3)

pq= (2⁴×3²)×(2³×3¹)

pq= (2⁴× 2³)× (3²×3¹)

pq= 2⁷ × 3³

pq = 3456

H.C.F. (pq) = 24 = 2³ × 3¹

L.C.M. depends on pq & H.C.F.

L.C.M = pq & H.C.F= 3456 & 24

(p,q) = (24, 144) ; (48, 72) ; (72, 48) ; (144. 24)

Hence, the two digit numbers are= 24 or, 48 or 72 whose L.C.M = 144 & H.C.F = 24

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Hope this will help you....