Let K be a positive integer such that k+4 is divisible by 7. Then the smallest positive integer n, greater than 2 , such that k+ 2n is divisible by 7 equals.?

Question

Answer

Since (k+4) is divisible by 7, so remainder when (k+4) is divided by 7 is 0.

k + 2n should be divisible by 7.

k + 2n + 4 - 4 should be divisible by 7.

k + 4 + 2n - 4 should be divisible by 7.

Given that k+4 is already divisible by 7. So, 2n-4 should be divisible by 7.

for example, n = 9 satisfies this condition.

Answer

Solution:

Given that k4 is divisible by 7.

We need to find the smallest positive integer n, greater than 2 , such that k 2n is divisible by 7.

Let us consider k = 1.

Then, k4 = 1.

As 1 is not divisible by 7, we need to consider k > 1.

Let us consider k = 2.

Then, k4 = 16.

As 16 is not divisible by 7, we need to consider k > 2.

Let us consider k = 3.

Then, k4 = 81.

As 81 is not divisible by 7, we need to consider k > 3.

Let us consider k = 4.

Then, k4 = 256.

As 256 is not divisible by 7, we need to consider k > 4.

Let us consider k = 5.

Then, k4 = 625.

As 625 is not divisible by 7, we need to consider k > 5.

Let us consider k = 6.

Then, k4 = 1296.

As 1296 is not divisible by 7, we need to consider k > 6.

Let us consider k = 7.

Then, k4 = 2401.

As 2401 is divisible by 7, we can take k = 7.

Now, we need to find the smallest positive integer n, greater than 2, such that k 2n is divisible by 7.

Let us consider n = 3.

Then, k 2n = 728.

As 728 is divisible by 7, we can take n = 3.

Hence, the smallest positive integer n, greater than 2, such that k 2n is divisible by 7 is 3.

Therefore, the answer is 3.

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