Prove-that-for-p-q-r-N-lcm-p-q-r-gcd-p-q-r-p-q-r-gcd-p-q-gcd-q-r-gcd-r-p-

Question Number 66275 by Rasheed.Sindhi last updated on 12/Aug/19

P r o v e t h a t f o r p, q, r \in N

\frac{lcm (p, q, r)}{\gcd (p, q, r)} = \frac{p \times q \times r}{\gcd (p, q) \times \gcd (q, r) \times \gcd (r, p)}

Commented by Prithwish sen last updated on 12/Aug/19

p = {nn}_{1} n_{2} x, q = {nn}_{2} n_{3} y, r = {nn}_{3} n_{1} z (let)

lcm (p, q, r) = {nn}_{1} n_{2} n_{3} xyz, \gcd (p, q, r) = n

\gcd (p, q) = {nn}_{2}, \gcd (q, r) = {nn}_{3}, \gcd (r, p) = {nn}_{1}

LHS = \frac{{nn}_{1} n_{2} n_{3} xyz}{n} = n_{1} n_{2} n_{3} xyz

RHS = \frac{n^{3} n_{1}^{2} n_{2}^{2} n_{3}^{2} xyz}{n^{3} n_{1} n_{2} n_{3}} = n_{1} n_{2} n_{3} xyz

LHS = RHS proved .

Commented by Rasheed.Sindhi last updated on 13/Aug/19

T h a n k s S i r!