Dividend Formula and Remainder Basics
The core relation: Dividend = Divisor × Quotient + Remainder. The remainder is always less than the divisor. To find a remainder quickly, you can break the dividend or use remainder properties: the remainder of a sum/product equals the operation on individual remainders. Example: remainder of (17 × 19) ÷ 5 = (remainder of 17÷5) × (remainder of 19÷5) = 2 × 4 = 8, and 8 ÷ 5 leaves 3. So the final remainder is 3. For large powers, find the repeating cycle of remainders. These shortcuts save time versus full long division.
Unit Digit (Cyclicity) Trick
The unit (last) digit of a power repeats in a cycle of at most 4. Digits 0, 1, 5, 6 always keep the same unit digit. For others, the cycle length is: 2 -> (2,4,8,6); 3 -> (3,9,7,1); 7 -> (7,9,3,1); 8 -> (8,4,2,6); these have cycle 4. Digits 4 -> (4,6) and 9 -> (9,1) have cycle 2. Method: divide the power by 4, look at the remainder. If remainder is 1,2,3 take that position in the cycle; if remainder is 0, take the 4th (last) value. Example: unit digit of 7^52 -> 52 ÷ 4 remainder 0 -> 4th value of (7,9,3,1) = 1.
Worked Example - Unit Digit of a Power
Q: Find the unit digit of 3^47. Solution: The unit digit cycle of 3 is (3, 9, 7, 1) repeating every 4 powers. Divide the exponent 47 by 4: 47 ÷ 4 = 11 remainder 3. A remainder of 3 means we take the 3rd value in the cycle, which is 7. So the unit digit of 3^47 is 7. (Quick check: 3^1=3, 3^2=9, 3^3=27 ends in 7 — matches the 3rd position.) Answer: 7.
Remainders and Unit Digits — Summary
Unit-digit and remainder questions are speed gold in SSC CHSL — one of the two usually appears, and both are pure pattern recognition. No calculation of huge powers needed; just cyclicity.
Unit-digit cyclicity (power's unit digit)
| Base ends in | Cycle | Period |
|---|---|---|
| 2 | 2,4,8,6 | 4 |
| 3 | 3,9,7,1 | 4 |
| 7 | 7,9,3,1 | 4 |
| 8 | 8,4,2,6 | 4 |
| 4 | 4,6 | 2 |
| 9 | 9,1 | 2 |
| 0,1,5,6 | same digit | 1 |
Method: take the exponent mod the period; a remainder of 0 means use the last digit of the cycle.
Remainders of big powers
- Cyclicity method: find the pattern of remainders and reduce the exponent mod the cycle length. 2^100 ÷ 7: powers give 2,4,1,2,4,1… (period 3); 100 mod 3 = 1 → remainder 2.
- Remainder splits over addition/multiplication: (a×b) mod n = [(a mod n)(b mod n)] mod n.
- Unit digit of a product/sum = unit digit of the product/sum of the unit digits.
Exam Tricks & Tips
- 🎯 Unit digit of 7^105: 105 mod 4 = 1 → first term of 7's cycle = 7.
- 🎯 If exponent mod period = 0, pick the last number in the cycle (e.g. 2^n with n≡0 → 6).
- 🎯 Unit digit of any number ending in 0,1,5,6 stays the same for every power.
- 🎯 For remainders, replace each factor by its remainder first, then multiply — keeps numbers tiny.
- 🎯 Unit digit of 4^odd = 4, 4^even = 6; 9^odd = 9, 9^even = 1 (period-2 shortcut).
- ❌ Don't reduce the exponent mod 10 for unit digits — reduce it mod the cycle length (usually 4).
Expected exam pattern
"Find the unit digit of 17^256 × 13^115" or "Remainder when 2^54 is divided by 7". Single pattern step, ~25–30 seconds.
Quick recap
Learn the four period-4 cycles (2,3,7,8) and the period-2 pairs (4,9); reduce the exponent mod the period. For remainders, use cyclicity and replace factors by their remainders before multiplying.
Remainders and Unit Digits — Flashcards
Cover the answer, recall, then check. 11 cards on unit digits & remainders for SSC CHSL.
Q1. Unit-digit cycle of powers of 2?
A1. 2, 4, 8, 6 (period 4).
Q2. Unit-digit cycle of powers of 3?
A2. 3, 9, 7, 1 (period 4).
Q3. Find the unit digit of 7^105.
A3. 105 mod 4 = 1 → first term of 7's cycle (7,9,3,1) = 7.
Q4. Unit digit of 2^100 — method and answer.
A4. 100 mod 4 = 0 → use the last term of 2,4,8,6 = 6.
Q5. Remainder when 2^54 is divided by 7?
A5. Powers of 2 mod 7 cycle 2,4,1 (period 3); 54 mod 3 = 0 → last term = 1.
Q6. Unit digit of any power of a number ending in 5 or 6?
A6. Stays 5 or 6 respectively — always the same digit.
Q7. Unit digit of 4 raised to an odd power vs an even power?
A7. Odd → 4, even → 6.
Q8. Unit digit of a product equals what?
A8. The unit digit of the product of the individual unit digits.
Q9. Unit digit of 13^115 × 17^256?
A9. 3^115: 115 mod 4 = 3 → 7. 7^256: 256 mod 4 = 0 → 1. 7×1 = 7.
Q10. When exponent mod period = 0, which cycle term do you take?
A10. The last term of the cycle.
Q11. (a×b) mod n equals?
A11. [(a mod n) × (b mod n)] mod n — reduce factors first, then multiply.