One trick you could do is factor out the GCF 7^14 and note how 50 is one of the factors left over
7^16 + 7^14 = 7^14(7^2 + 1)
7^16 + 7^14 = 7^14(49 + 1)
7^16 + 7^14 = 7^14*50
Since 50 is a factor of the original numeric expression, this means the original expression is divisible by 50.
