For what least value of n,![Screenshot 2025-05-23 162520.png](https://res.cloudinary.com/aptidude/image/upload/v1747997730/aptidude_questions/rhh8afqsir6ivalzf725.png) is completely divisible by 6, where n is an integer?

Question

For what least value of n,![Screenshot 2025-05-23 162520.png](https://res.cloudinary.com/aptidude/image/upload/v1747997730/aptidude_questions/rhh8afqsir6ivalzf725.png)  is completely divisible by 6, where n is an integer?

Accepted Answer

We need to find the least integer n such that 2^(2n) + 2n is divisible by 6.

Step 1: Understand divisibility by 6 means divisibility by both 2 and 3.

Step 2: Check divisibility by 2:
- 2^(2n) is always even (since 2 raised to any positive power is even).
- 2n is also even (since 2 times any integer is even).
- Sum of two even numbers is even, so the expression is always divisible by 2.

Step 3: Check divisibility by 3:
We want (2^(2n) + 2n) ≡ 0 (mod 3).

Step 4: Find pattern of 2^(2n) mod 3:
- 2^1 mod 3 = 2
- 2^2 mod 3 = 4 mod 3 = 1
- 2^3 mod 3 = 2
- 2^4 mod 3 = 1
So powers of 2 mod 3 cycle every 2 powers: 2,1,2,1,...

Since exponent is 2n (even), 2^(2n) mod 3 = 1 for all n.

Step 5: So expression mod 3 is:
(2^(2n) + 2n) mod 3 = (1 + 2n) mod 3 = 0
=> 2n ≡ -1 ≡ 2 (mod 3)
=> 2n ≡ 2 (mod 3)

Step 6: Multiply both sides by the modular inverse of 2 mod 3.
Since 2 * 2 = 4 ≡ 1 mod 3, inverse of 2 mod 3 is 2.

Multiply both sides by 2:
2 * 2n ≡ 2 * 2 (mod 3)
=> 4n ≡ 4 (mod 3)
=> 1n ≡ 1 (mod 3) (since 4 mod 3 = 1)

So n ≡ 1 (mod 3)

Step 7: Least non-negative integer n satisfying this is n=1.

Step 8: Check n=1:
2^(2*1) + 2*1 = 2^2 + 2 = 4 + 2 = 6, which is divisible by 6.

Step 9: Check n=0:
2^(0) + 0 = 1 + 0 = 1, not divisible by 6.

Step 10: Check n=2:
2^(4) + 4 = 16 + 4 = 20, not divisible by 6.

Step 11: Check n=3:
2^(6) + 6 = 64 + 6 = 70, not divisible by 6.

Wait, 70 mod 6 = 4, not divisible.

Re-examine step 6:
We found n ≡ 1 mod 3, so n = 1,4,7,...

Check n=4:
2^(8) + 8 = 256 + 8 = 264
264 mod 6 = 0 (since 6*44=264)

So n=1 works, n=4 works.

Since n=1 is the least integer satisfying the condition, the answer is 1.

Final answer: 1

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