Question

Let a and b be two positive integers such that a = p³q⁴ and b = p²q³, where p and q are prime numbers. If HCF (a, b) = p^mqⁿ and LCM (a, b) = p^rq^s, then (m + n)(r + s) = ______.

पर्याय

• 15

• 30

• 35

• 72

Answer 1

Let a and b be two positive integers such that a = p³q⁴ and b = p²q³, where p and q are prime numbers. If HCF (a, b) = p^mqⁿ and LCM (a, b) = p^rq^s, then (m + n)(r + s) = 35.

Explanation:

Given two numbers

a = p³q⁴ and b = p²q³

p	p3q4
p	p2q4
p	pq4
p	q4
q	q3
q	q2
q	q
	1

 

p	p2q3
p	pq3
q	q3
q	q2
q	1

Finding HCF

a = p³q⁴ = p × p × p × q × q × q × q

b = p²q³ = p × p × q × q × q

HCF = p × p × q × q × q

HCF = p²q³

Comparing HCF = p²q³ with HCF = p^mqⁿ

∴ m = 2, n = 3

p	p3q4, p2q3
p	p2q4, p2q3
p	pq4, q3
q	q4, q3
q	q3, q2
q	q2, q
q	q, 1
	1, 1

Finding LCM

LCM = p × p × p × q × q × q × q

LCM = p³q⁴

Comparing LCM = p³q⁴ with LCM = p^rq^s

∴ r = 3, s = 4

Now, (m + n)(r + s) = (2 + 3) × (3 + 4)

= 5 × 7

= 35