72n+16n-1 is a multiple of 64: The expression 72n + 16n – 1 is a multiple of 64 if and only if (72 + 16)n – 1 is a multiple of 64. Simplifying, 88n – 1 is a multiple of 64, so n must satisfy the equation 88n – 1 = 64k, where k is an integer. Dividing both sides of the equation by 88, we get n = 64k/88 + 1/88. Since n must be an integer, it follows that 64k must be a multiple of 88. Thus, the expression is a multiple of 64 if and only if k is a multiple of 11.
If you want more information about the relationship between 64 and 88, you can observe that they are both divisible by the same prime factors. The prime factorization of 64 is 2^6, and the prime factorization of 88 is 2^2 * 11. So, 64 and 88 share the factor of 2^2, and since 64 is a multiple of 2^2, it is also a multiple of 88.
Therefore, any multiple of 64 is also a multiple of 88 and vice versa. In other words, if an expression is a multiple of 64, it is automatically a multiple of 88, and this is why the expression 72n + 16n – 1 is a multiple of 64 if and only if (72 + 16)n – 1 is a multiple of 64.