How to Solve Remainders Problems (GRE/GMAT) – Complete Guide

How to Solve Remainders Problems (GRE/GMAT) – Complete Guide

Score Impact: Remainders problems appear 3-5 times per GRE and 4-6 times per GMAT—mastering them can boost your Quant score by 5-7 points, moving you from the 70th to the 90th percentile.

WHAT THIS QUESTION TYPE IS ACTUALLY TESTING

The GRE/GMAT isn’t testing your ability to compute remainders—it’s testing: 1. Pattern recognition – Can you spot cycles in remainders (e.g., powers of 2, 3, 5)? 2. Algebraic translation – Can you convert a word problem into a modular equation (N ≡ r mod d)? 3. Elimination under pressure – Can you quickly rule out wrong answers without full computation?

ANATOMY OF THE QUESTION

Structure Breakdown

1. Stem: A scenario involving division, cycles, or modular arithmetic.
2. Example: "When positive integer N is divided by 5, the remainder is 3. What is the remainder when 2N is divided by 5?"
3. Conditions: Constraints on N (e.g., N is a two-digit number, N is odd).
4. Answer Choices: Usually 5 options (A-E), with 1-2 traps based on misapplying remainder rules.
5. What to Ignore: Overcomplicating the problem (e.g., solving for N when only the remainder is needed).

Representative Example

When positive integer N is divided by 6, the remainder is 4. What is the remainder when 3N is divided by 6? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4

THE DECISION FRAMEWORK (Step-by-Step)

Run this process every time. No exceptions.

1. Translate the stem into a modular equation.
2. N ≡ r mod d means N = d·k + r, where 0 ≤ r < d.

3. Example: N ≡ 4 mod 6 → N = 6k + 4.

4. Express the target expression in terms of the given remainder.

5. Multiply, add, or exponentiate N as needed.

6. Example: 3N = 3(6k + 4) = 18k + 12.

7. Simplify the expression modulo d.

8. Break it into d·(multiple) + remainder.

9. Example: 18k + 12 = 6(3k + 2) + 0 → remainder is 0.

10. Check for cycles or shortcuts.

11. If the problem involves exponents (e.g., N²), test small values of N to spot patterns.

12. Example: If N ≡ 2 mod 5, then N² ≡ 4 mod 5, N³ ≡ 3 mod 5, etc.

13. Eliminate wrong answers.

14. Use the remainder to cross out options that violate 0 ≤ r < d.

15. Example: If remainder is 0, eliminate (B)-(E).

16. Verify with a concrete number (if unsure).

17. Pick a number satisfying the condition (e.g., N = 10 for N ≡ 4 mod 6).
18. Compute the target expression (3×10 = 30) and divide by 6 → remainder 0.

Worked Examples

Example 1 – Straightforward

When N is divided by 7, the remainder is 5. What is the remainder when 4N is divided by 7? (A) 1 (B) 2 (C) 3 (D) 4 (E) 6

Step-by-Step: 1. N ≡ 5 mod 7 → N = 7k + 5. 2. 4N = 4(7k + 5) = 28k + 20. 3. 28k ≡ 0 mod 7 → 20 mod 7 = 6 (since 7×2=14, 20-14=6). 4. Remainder is 6 → Answer: (E).

Elimination: - (A)-(D) are too small; only (E) fits 0 ≤ r < 7.

Example 2 – Common Trap Version

When N is divided by 8, the remainder is 3. What is the remainder when N² is divided by 8? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5

Trap: Students compute N = 8k + 3 → N² = 64k² + 48k + 9 → 9 mod 8 = 1 → (A). But: The trap is assuming k=0 (i.e., N=3). What if k=1 (N=11)? - 11² = 121 → 121 mod 8 = 1 (since 8×15=120, 121-120=1). Answer: (A).

Why the trap fails: - The remainder is consistent for all k, but students might doubt and pick (C) (the original remainder).

Example 3 – Hard Variant

When N is divided by 5, the remainder is 2. What is the remainder when N³ + 2N is divided by 5? (A) 0 (B) 1 (C) 2 (D) 3 (E) 4

Step-by-Step: 1. N ≡ 2 mod 5 → N = 5k + 2. 2. Compute N³ and 2N separately:
- N³ ≡ 2³ ≡ 8 ≡ 3 mod 5 (since 8-5=3).
- 2N ≡ 2×2 ≡ 4 mod 5. 3. N³ + 2N ≡ 3 + 4 ≡ 7 ≡ 2 mod 5. 4. Remainder is 2 → Answer: (C).

Elimination: - (A) 0: Too large a jump from 2. - (B) 1: Doesn’t match 3+4=7. - (D) 3: Only N³’s remainder, not the sum. - (E) 4: Only 2N’s remainder.

WRONG ANSWER PATTERNS

1. Original Remainder Trap
2. Why it looks right: Students assume the remainder stays the same after operations.
3. Why it’s wrong: Operations (e.g., 2N, N²) change the remainder.

4. Example: If N ≡ 3 mod 5, 2N ≡ 1 mod 5 (not 3).

5. Off-by-One Error

6. Why it looks right: Students misapply N = d·k + r (e.g., r ≥ d).
7. Why it’s wrong: Remainders must satisfy 0 ≤ r < d.

8. Example: N ≡ 5 mod 4 is invalid; correct is N ≡ 1 mod 4.

9. Partial Computation

10. Why it looks right: Students compute only part of the expression (e.g., N² but not N² + N).
11. Why it’s wrong: The question asks for the full expression’s remainder.

12. Example: For N² + N, computing only N² gives the wrong answer.

13. Cycle Ignorance

14. Why it looks right: Students forget that remainders repeat in cycles (e.g., powers of 2 mod 5: 2, 4, 3, 1, 2, ...).
15. Why it’s wrong: Leads to incorrect assumptions about higher powers.
16. Example: 2⁴ ≡ 1 mod 5, not 16 mod 5 = 1.

Common Mistakes

1. Mistake: Solving for N instead of the remainder.
2. Why it happens: Overcomplicating the problem.

3. Correct approach: Use modular arithmetic to isolate the remainder.

4. Mistake: Forgetting 0 ≤ r < d.

5. Why it happens: Carelessness with definitions.

6. Correct approach: Always check if r is within bounds.

7. Mistake: Assuming N is the smallest possible number.

8. Why it happens: Testing only N = r (e.g., N=3 for N ≡ 3 mod 5).

9. Correct approach: Test N = d + r (e.g., N=8) to confirm patterns.

10. Mistake: Misapplying exponent rules.

11. Why it happens: Confusing (a + b)² with a² + b².

12. Correct approach: Expand fully or use modular shortcuts (e.g., N ≡ 2 mod 5 → N² ≡ 4 mod 5).

13. Mistake: Skipping elimination.

14. Why it happens: Rushing to compute.
15. Correct approach: Cross out impossible answers first (e.g., remainders ≥ d).

TIME STRATEGY

• Target time: 1:15 per question.
• When to skip: If you can’t translate the stem into a modular equation in 20 seconds.
• Minimum work:
• Write N ≡ r mod d.
• Express the target in terms of N.
• Simplify modulo d.
• Eliminate 2-3 wrong answers.

BACKSOLVING AND SHORTCUTS

1. Pick Numbers:
2. Choose N satisfying the condition (e.g., N=7 for N ≡ 2 mod 5).

3. Compute the target expression and divide by d to find the remainder.

4. Cycle Shortcuts:

5. For powers, list remainders until a cycle appears:

   • 2ⁿ mod 5: 2, 4, 3, 1, 2, 4, 3, 1, ...
   • 3ⁿ mod 7: 3, 2, 6, 4, 5, 1, 3, ...

6. Elimination-First:

7. If the remainder must be even, cross out odd options.
8. If d=5, eliminate options ≥5.

1-Minute Recap

"Here’s your 3-step process for remainders problems: 1. Translate: Write N ≡ r mod d from the stem. 2. Express: Multiply, add, or exponentiate N as needed. 3. Simplify: Break it into d·(multiple) + remainder and find r.

Traps to avoid: - Don’t assume the remainder stays the same after operations. - Always check 0 ≤ r < d. - Test a concrete number if you’re stuck.

Time hack: If you’re over 90 seconds, pick a number and backsolve. Now go crush these questions!

Final Note: Remainders problems are predictable. Follow the framework, and you’ll solve them faster than the clock.