Math, asked by josephsebinb, 2 months ago

A pair of integers( m, n) satisfy m × n = to gcd (m, n) +1cm(m,n) then m+n is​

Answers

Answered by RvChaudharY50
0

Given :- A pair of integers( m, n) satisfy m × n = to gcd (m, n) + lcm(m,n) then m+n is ?

Solution :-

we know that, for two number a and b

  • a * b = LCM of a and b * GCD of a and b .
  • LCM is always divisible by HCF .

so, for 2 number m and n , given that,

→ m * n = gcd(m,n) + lcm(m,n)

and , we know that,

→ m * n = gcd(m,n) * lcm(m,n)

comparing value of m * n , we get,

→ gcd(m,n) + lcm(m,n) = gcd(m,n) * lcm(m,n)

now, since lcm is multiple of gcd(m,n) or we can say

  • lcm(m,n) = M * gcd(m,n) { where M is any natural number. }

putting this value in both sides we get,

→ gcd(m,n) + M * gcd(m,n) = gcd(m,n) * M * gcd(m,n)

→ gcd(m,n)[1 + M] = gcd(m,n) * M * gcd(m,n)

→ [1 + M] = M * gcd(m,n)

→ [1 + M]/M = gcd(m,n)

given that, m and n both are integers . then, M must be equal to 1 .

→ 2 = gcd(m,n)

therefore,

→ lcm(m,n) = M * gcd(m,n)

→ lcm(m,n) = 1 * 2

→ lcm(m,n) = 2

as,

→ lcm(m,n) = gcd(m,n)

→ m = n = 2 .

hence,

→ m + n = 2 + 2 = 4 (Ans.)

Learn more :-

let a and b positive integers such that 90 less than a+b less than 99 and 0.9 less than a/b less than 0.91. Find ab/46

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