Remainders & Modular Reasoning
Remainders look intimidating because the basic on-screen calculator will not hand you one directly — it only gives a decimal. That gap is exactly why the GRE likes them: the questions reward reasoning over button-pressing. The core identity is simple, dividend = divisor × quotient + remainder, with the remainder always between 0 and one less than the divisor. From there, a few pattern tools — the cyclic behaviour of units digits, and how remainders combine under addition and multiplication — let you answer "what is the units digit of 7⁴⁵?" or "what is the remainder when a huge product is divided by 5?" in seconds. This chapter turns those patterns into reflexes.
Topics
⚡ GRE shortcuts & speed methods
The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.
- The calculator gives a decimal, not a remainder. Convert: remainder = N − divisor × (whole-number quotient).
- Reduce before you compute. Replace each number by its remainder first, then add or multiply — it keeps everything mentally small.
- Units-digit cycles are all length 1, 2 or 4. Take the exponent mod the cycle length and read the position.
- Use negative remainders to simplify: 2 ≡ −1 (mod 3) turns 2^n into (−1)^n instantly.
- Same remainder for several divisors ⇒ answer is LCM(divisors)·k + r.
- Remainders that are all one short of their divisor ⇒ answer is LCM(divisors)·k − 1.
⚠️ Common mistakes & traps
GRE is designed so that careless errors here cost you marks. Internalise each trap before the exam.
- Reporting a remainder equal to or larger than the divisor — it must be strictly less.
- Reading the calculator's decimal part as the remainder (0.4 is not remainder 4).
- Multiplying big numbers before reducing, instead of reducing to remainders first.
- Using the wrong cycle length for units digits (assuming everything is a 4-cycle; 4 and 9 are 2-cycles).
- Treating "remainder 0" as "no remainder / undefined" — remainder 0 just means it divides evenly.
- Forgetting the shared-remainder shortcut and grinding through lists when LCM·k + r would answer instantly.
📈 GRE exam insight & question patterns
Quantitative Comparison — Quantity A: the units digit of 3⁴⁴. Quantity B: the units digit of 2⁴⁴. Which is greater?
3 cycles 3,9,7,1: 44 ≡ 0 mod 4 → 1. 2 cycles 2,4,8,6: 44 ≡ 0 → 6. Quantity B is greater.
Numeric Entry — enter the remainder when 25 × 33 × 41 is divided by 4.
Reduce mod 4: 25 ≡ 1, 33 ≡ 1, 41 ≡ 1, so the product ≡ 1. Remainder is 1.
A number leaves the same remainder 1 when divided by 2, 3, 4 and 6. What is the smallest such number above 1?
LCM(2,3,4,6) = 12, so it is 12k + 1; the smallest above 1 is 13.
Multiple-answer — which of 6, 10, 16, 22, 28 leave remainder 4 on division by 6? Select all.
16 = 6·2+4 and 22 = 6·3+4 and 28 = 6·4+4 all leave 4; 6 leaves 0 and 10 leaves 4 as well (10 = 6·1+4). So 10, 16, 22, 28.
🎴 Flashcards — instant recall
Tap a card to reveal the answer. Drill these until they are automatic.
📌 Quick revision
- Every division obeys N = d·q + r with the remainder strictly less than the divisor.
- The basic calculator gives decimals, not remainders — convert deliberately.
- Units digits of powers cycle in lengths 1, 2 or 4; reduce the exponent mod the cycle length.
- Remainders add and multiply, so reduce numbers to their remainders before computing.
- Negative remainders (like 2 ≡ −1 mod 3) collapse large powers instantly.
- A shared remainder across divisors gives LCM·k + r; all-one-short gives LCM·k − 1.
- Divisor–remainder word problems reduce to the division identity plus an LCM.
- Remainder 0 is legitimate — it simply means the divisor goes in evenly.
Chapter test
🏆 Vidaara GRE success checklist
You have truly mastered Remainders & Modular Reasoning when you can tick every box below.
- Recall every formula in this chapter without looking them up
- Solve each topic’s practice set with at least 80% accuracy
- Use the chapter shortcuts to cut your solving time in half
- Spot and avoid every common trap listed above
- Score 80%+ on the timed chapter test
📋 Chapter mastery scorecard
Track where you stand. Aim for the target before moving to the next chapter.
| Skill checkpoint | Target |
|---|---|
| Concept theory & formulas understood | 100% |
| Topic practice sets attempted (4 topics) | 4/4 |
| Best topic-test score | — → 80%+ |
| Chapter test score | — → 80%+ |
| Flashcards drilled to instant recall | 8 cards |
Formula Reference Sheet
Division & remainders
| Division identity | N = d·q + r, 0 ≤ r < d |
|---|---|
| Remainder range | 0 to (divisor − 1) |
| Remainder of a sum | (r₁ + r₂) mod d |
| Remainder of a product | (r₁ × r₂) mod d |
Units-digit cycles
| Cycle of 2 | 2, 4, 8, 6 (length 4) |
|---|---|
| Cycle of 3 | 3, 9, 7, 1 (length 4) |
| Cycle of 7 | 7, 9, 3, 1 (length 4) |
| Cycle of 8 | 8, 4, 2, 6 (length 4) |
| Stable digits | 0,1,5,6 repeat unchanged |