Question

Find two digit numbers having gcf 24 and lcm 144

Answer 1

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

==================================================================
Hope this will help you....

Answer 2

We are given in the question that L.C.M is equal to  144 and  

Also, H.C.F.is equal to 24
Let us  consider we have the two numbers be k and  j respectively.As we know that
L.C.M*H.C.F = Product of the two numbers
L.C.M. = (k*j) / H.C.F.
Also 144 = kj / 24
Thus , kj= 144× 24
And also we can write it as kj= (2⁴×3²)×(2³×3¹)
Thus , this will become ,kj= 2⁷ × 3³ = 3456 
And also ,H.C.F. (kj) =  2³ × 3¹= 24
SO now we know L.C.M. depends on the kj and also on  H.C.F. 
Thus it will become like that  L.C.M = kj and also H.C.F is equal to 3456 and 24 
so we can say that (k,j) =(24, 144);(48, 72);(72, 48);(144, 24)
so here
The digit will be like the 24 , 48 , 72 which have L.C.M = 144 and H.C.F = 24

Hope this will helps.