How many distinct remainders are possible when 16^n is divided by 9, where n is any positive integer?

Introduction / Context:
This question checks knowledge of remainders (also called residues) in modular arithmetic, especially for powers of a number. Instead of directly computing large powers, we look for patterns in remainders when 16^n is divided by 9. Recognising cyclic patterns is a key skill in number theory and is very useful in many aptitude exams.

Given Data / Assumptions:

• We consider 16^n for positive integers n (n = 1, 2, 3, ...).
• Each such value is divided by 9.
• We want to know how many different remainders can appear.
• Remainders when dividing by 9 must be between 0 and 8 inclusive.

Concept / Approach:
Instead of calculating huge powers, we use modular arithmetic. First reduce the base modulo 9, then compute successive powers of this reduced base and observe the pattern in residues. If the sequence of remainders repeats after some point, we have found a cycle, and the number of distinct remainders is the length of this cycle. This method avoids large computations and is a standard approach in such problems.

Step-by-Step Solution:
Step 1: Reduce the base 16 modulo 9. Since 16 = 9 + 7, we have 16 ≡ 7 (mod 9).Step 2: Therefore 16^n ≡ 7^n (mod 9) for all positive integers n.Step 3: Compute the first few powers of 7 modulo 9.Step 4: For n = 1, 7^1 = 7, so the remainder is 7.Step 5: For n = 2, 7^2 = 49. Dividing 49 by 9 gives remainder 49 - 45 = 4.Step 6: For n = 3, 7^3 = 7 * 49 = 343. Dividing 343 by 9 gives remainder 343 - 9 * 38 = 343 - 342 = 1.Step 7: For n = 4, 7^4 = 7 * 343 = 2401. Since 7^3 ≡ 1 (mod 9), 7^4 ≡ 7 * 1 = 7 (mod 9).Step 8: We now see a cycle in remainders: 7, 4, 1, then 7 again.Step 9: Because the pattern repeats every 3 powers, the distinct remainders are 7, 4, and 1.Step 10: Thus the number of distinct remainders possible is 3.

Verification / Alternative check:
You can continue the pattern to be sure. For n = 5, 7^5 ≡ 7^2 ≡ 4 (mod 9), and for n = 6, 7^6 ≡ 7^3 ≡ 1 (mod 9). The cycle 7, 4, 1 clearly repeats every three steps. Since modular arithmetic with a fixed modulus cannot create new residues once a cycle is detected, we are confident that only 7, 4, and 1 occur as remainders of 16^n divided by 9 for positive n.

Why Other Options Are Wrong:
Option 1: This would mean the remainder is always the same, which is clearly not true because 16^1 and 16^2 give different remainders modulo 9.Option 2: This underestimates the cycle; we have already seen three distinct remainders: 7, 4, and 1.Option 4: This overestimates the number of distinct remainders; the cycle length is 3, not 4.Option 6: This is far too large for a modulus 9 question where we have explicitly identified only three residues in the cycle.

Common Pitfalls:
Students sometimes forget to reduce the base modulo the divisor and instead try to compute huge powers directly, which is impractical. Another mistake is to stop too soon and assume a pattern after only two values. It is important to verify that a cycle truly repeats. Confusing 16^n with 16 * n is also a common misreading. Always read the expression carefully and remember that powers behave very differently from simple multiplication when working with remainders.

Final Answer:
The number of distinct remainders possible is 3.