some short and selected math problems of different levels in random order I try to keep the ans simple
Saturday, March 29, 2014
2014/027) Solve in natural numbers 2^x – 3^y = 7
x cannot be odd because if x is odd then
2^x mod 3 = -1 so 2^x – 3^y mod 3 = -1 so it cannot be 7 as 7 = 1 mod 3
y cannot be odd as if y is odd 3^y =3 mod 8
so 2^x – 3y = 5 mod 8 for x >= 3
if x = 1 or 2 2^ x < 7 so 2^x – 3^y = 7 not possible
so x and y both are even
say x = 2a and y = 2b
so 2^2a – 3^2b = 7
or (2^a + 3^b)(2^a- 3^b) = 7
so 2^a + 3^b = 7 and 2^a – 3^b = 1
solving these 2 we get a = 2 and b = 1 or x= 4 and y = 2
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