Find all pair of integers ab with 1 a 25 1 b 25 i for which

Find all pair of integers (a,b) with 1 a 25, 1 b 25

i. for which gcd(a,b) = 2

ii. for which lcm(a,b) = 210

iii. for which both gcd(a,b) = 2 and lcm.(a,b) = 210

iv. What can be said in general about a pair of integers if their gcd is 2?

Thank you

Solution

we are given that

1 a 25, 1 b 25

(i)

We have to find pair of integers

such that gcd(a,b)=2

(a,b)=(2,4) (2,6) (2,8) (2,10) (2,12) (2,14) (2,16) (2,18) (2,20) (2,22) (2,24) (4,6) (4,10) (4,14) (4,18) (4,22) (6,4) (6,8) (6,10) (6,14) (6,20) (6,22) (8,10) (8,14) (8,18) (8,22) (10,12) (10,14) (10,16) (10,18) (10,22) (12,14) (12,22) (14,16) (14, 18) (14,20) (14,22) (14, 24) (16,18) (16,22) (18,20) (18,22) (20,22) (22,24)

(ii)

We have to find for lcm(a,b)=210

210=2*3*5*7

possible integers as (10, 21) (14, 15)

(iii)

Possible integer for lcm(a,b)= (10, 21) (14, 15)

and we can see that gcd(a,b)=2

none of them are common

so, no pair of such integers are possible

(iv)

if gcd=2

It means that common factor between a and b will be \'2\'