Question

Find the hcf and lum of 65 and 117 and express it in the form of 65m+117n

Answer 1

Given integers,
117 and 65
Applying Euclid's division lemma we get,
117=65×1+52.............................................1
Applying Euclid's division lemma to 65 and 52 we get
65=52×1+13..............................................2
Applying Euclid's division lemma to 52 and 13 we get,
52=13×4+0................................................3
The remainder at this stage is zero.
Hence HCF of 117 and 65 is 13.
eq1 and eq2 can also be written as,
117-65×1=52.............................................. 4

65-52×1=13................................................5
Eq5 we have,
13=65-52×1
By putting eq4 in above equation we get,
13=65-(117-65×1)1

13=65-117×1+65×1

13=65×2-117×1

13=65(2)+117(-1)

13=65m+117n , where m=2 and n=-1

Hence HCF of 117 and 65 is in the form of 65m+117n such that m=2 and n=-1.

Prime factorization of,
117=3^2×13
65=5×13
LCM=product of each prime factor of highest power
LCM=3^2×5×13
LCM=585.
Hence LCM of 117 and 65 is 585.