Question

Let a and b are positive integers and if a=bq+r and 0 (smaller than/equal to) r<b, then:
(a)LCM(a,b)=LCM(b,r)
(b)LCM(a,b)=HCF(b,r)
(c) HCF(a,b)=HCF(b,r)
(d)HCF(a,b)=LCM(b,r)​

Answer 1

Answer:

Step-by-step explanation:

SOLUTION

GIVEN

a and b are positive integers, then you know that a = bq + r, such that 0 ≤ r ≤ b , where q is a whole number.

TO PROVE

HCF(a,b) = HCF(b, r)

PROOF

Let c = HCF(a,b) & d = HCF(b, r)

Since c = HCF(a,b)

⟹ c divides a and c divides b

⟹ c divides a and c divides bq

⟹ c divides a - bq

⟹ c divides r

⟹ c is a common divisor of b & r

⟹ c divides d

Similarly we can show that d divides c

Now c and d are positive integers

Consequently c = d

Hence HCF(a,b) = HCF(b, r)

Hence proved

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GIVEN

a and b are positive integers, then you know that a = bq + r, such that 0 ≤ r ≤ b , where q is a whole number.

TO PROVE

HCF(a,b) = HCF(b, r)

PROOF

Let c = HCF(a,b) & d = HCF(b, r)

Since c = HCF(a,b)

⟹ c divides a and c divides b

⟹ c divides a and c divides bq

⟹ c divides a - bq

⟹ c divides r

⟹ c is a common divisor of b & r

⟹ c divides d

Similarly we can show that d divides c

Now c and d are positive integers

Consequently c = d

Hence HCF(a,b) = HCF(b, r)

Hence proved

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