1Plz solve the last one too.
1.Prove that 24 divides 2.7^n+3.5^n-5.(n>=1)
2.Prove that the sum of the squares of two odd integers cannot be a perfect square.
3.If there exists x and y for which ax+by=gcd(a,b),then gcd(x,y)=1.(a,b are integers)
4.Prove that for any odd a ,prove that gcd(3a,3a+2)=1.
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5 Answers
62First one is simple by induction....
Or by taking it as 2(7^n-1)+3(5^n-1) where 7^n-1 is a multiple of 24 and same with 5^n-1
62second one is by observation that if the first is an odd perfect square it is of the form 4n+1 while the other of the form 4m+1
Hence the sum will be 4t+2
Wile any perfect square will be a multiple of 4 whenever even.. Hence not possible
62The minimum values of ax+by is the gcd of a and b..
So if x and y themselves have a common factor, then we can look at the value of ax/k+by/k which will be smaller than ax+by hence a contradiction.
341If a prime p divides 3a and 3a+2, it must divide their difference i.e. 2. So p =2. But 2 does not divide 3a as 3a is odd. So, the gcd =1