find thr greatest values of p and q so that the even number 4pq4p is divisible by both 3 and 5
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We can test for the first few values of n (n=0, n=1, n=2, n=3, n=4...) and see a pattern emerging. It seems that when n is odd, 2%5En%2B1 is divisible by 3.
We can use induction to prove that this is true for all even values of n. We have tried base cases, so it holds for the base cases. Since any even number can be written as 2k where k is an integer, we will assume that for all values up to k, 2^{2*k}+1=3*m (m is an integer, meaning it is divisble by 3. This can be rewritten as:2^{2(k)}=3m-1
We must now prove that 2^{2(k+1)}+1 is divisible by 3.2^{2(k+1)}+1=2^(2k)*2^2+1=4*(3m-1)+1=12m-4+1=12m-3Now since both 12 and -3 are divisible by 3, then this whole value is divisible by three.. so our proof is done
We can use induction to prove that this is true for all even values of n. We have tried base cases, so it holds for the base cases. Since any even number can be written as 2k where k is an integer, we will assume that for all values up to k, 2^{2*k}+1=3*m (m is an integer, meaning it is divisble by 3. This can be rewritten as:2^{2(k)}=3m-1
We must now prove that 2^{2(k+1)}+1 is divisible by 3.2^{2(k+1)}+1=2^(2k)*2^2+1=4*(3m-1)+1=12m-4+1=12m-3Now since both 12 and -3 are divisible by 3, then this whole value is divisible by three.. so our proof is done
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