Number theory problem- 22 Dec

Question

If !!3^x 5^y = 75!! and !!5^x 3^y = 45!!

Then !!3(y-x)!!

!!a)" "6!!

!!b)" "3!!

!!c)" "0!!

!!a)" "!!None

Solution

Given equation !!3^x 5^y = 75!! and !!5^x 3^y = 45!!

Taking !!log!! both sides, we get

!!log3^x + log5^y = log75!!

!!=> xlog3 + ylog5 = log(25xx3)!!

!!=> xlog3 + ylog5 = log25 + log3)!!

!!=> xlog3 + ylog5 = 2log5 + log3)!!

Similary,

!!log5^x + log3^y = log45!!

!!=> xlog5 + ylog3 = log(9xx5)!!

!!=> xlog5 + ylog3 = log9 + log5!!

!!=> xlog5 + ylog3 = log5 + 2log3!!

Hence, equating both the equations we get

!!x = 1!! and !!y = 2!!

So, !!3(y-x) = 3(2-1) = 3!!