SSC CGL Maths: Unit Digit & Last 2 Digits – Complete Guide (Cyclicity Tricks)
๐ Series Info: Ye post Chapter 1: Number System ka part hai.
Post Number: 7/76 | Topic: 1.7 Unit Digit & Last 2 Digits
Previous: 1.1-1.6 Complete ✅
Post Number: 7/76 | Topic: 1.7 Unit Digit & Last 2 Digits
Previous: 1.1-1.6 Complete ✅
Introduction: Unit Digit Aur Last 2 Digits Kyu Important Hain?
Namaste dosto! Aaj ham Unit Digit (เคเคाเค เค ंเค) aur Last 2 Digits padhenge. Ye SSC CGL me high-speed calculation ke liye bahut important hai.
๐น Unit Digit = Number ka sabse right-side wala digit
๐น Last 2 Digits = Number ke right side ke 2 digits
๐น Last 2 Digits = Number ke right side ke 2 digits
Examples:
3456 ka unit digit = 6
3456 ka last 2 digits = 56
789 ka unit digit = 9
789 ka last 2 digits = 89
3456 ka unit digit = 6
3456 ka last 2 digits = 56
789 ka unit digit = 9
789 ka last 2 digits = 89
๐ฏ Exam Importance: SSC CGL me direct questions aate hain:
- Large powers ka unit digit (เคैเคธे: 7²⁰²⁵)
- Products ka unit digit
- Factorial ka unit digit
- Last 2 digits of powers
- Fast calculation without full calculation
Cyclicity Kya Hai? CORE CONCEPT
Cyclicity = Kisi number ke successive powers ke unit digits ka repeating pattern
Har digit (0-9) ka apna cyclicity hota hai. Isse samajh ke hum kisi bhi power ka unit digit instantly nikal sakte hain!
Cyclicity Table (Sabse Important!)
| Digit | Powers Ka Unit Digit Pattern | Cyclicity |
|---|---|---|
| 0 | 0, 0, 0, 0... | 1 |
| 1 | 1, 1, 1, 1... | 1 |
| 2 | 2, 4, 8, 6, 2, 4, 8, 6... | 4 |
| 3 | 3, 9, 7, 1, 3, 9, 7, 1... | 4 |
| 4 | 4, 6, 4, 6... | 2 |
| 5 | 5, 5, 5, 5... | 1 |
| 6 | 6, 6, 6, 6... | 1 |
| 7 | 7, 9, 3, 1, 7, 9, 3, 1... | 4 |
| 8 | 8, 4, 2, 6, 8, 4, 2, 6... | 4 |
| 9 | 9, 1, 9, 1... | 2 |
๐ก Memory Trick:
Cyclicity 1: 0, 1, 5, 6 (Hamesha same unit digit)
Cyclicity 2: 4, 9 (Alternate karte hain)
Cyclicity 4: 2, 3, 7, 8 (4 steps me repeat)
Cyclicity 1: 0, 1, 5, 6 (Hamesha same unit digit)
Cyclicity 2: 4, 9 (Alternate karte hain)
Cyclicity 4: 2, 3, 7, 8 (4 steps me repeat)
Cyclicity Use Karke Unit Digit Kaise Nikale?
Steps:
- Number ka unit digit dekho
- Us digit ki cyclicity check karo
- Power ko cyclicity se divide karo, remainder nikalo
- Remainder ke position ka unit digit answer hai
Example 1: 7²⁵ ka unit digit kya hoga?
Step 1: Unit digit = 7
Step 2: 7 ki cyclicity = 4 (pattern: 7, 9, 3, 1)
Step 3: 25 ÷ 4 = Quotient 6, Remainder 1
Step 4: Remainder = 1, toh 1st position = 7
Answer: Unit digit = 7 ✓
Step 1: Unit digit = 7
Step 2: 7 ki cyclicity = 4 (pattern: 7, 9, 3, 1)
Step 3: 25 ÷ 4 = Quotient 6, Remainder 1
Step 4: Remainder = 1, toh 1st position = 7
Answer: Unit digit = 7 ✓
Example 2: 3⁴⁷ ka unit digit?
Unit digit = 3
Cyclicity = 4 (pattern: 3, 9, 7, 1)
47 ÷ 4 = Remainder 3
3rd position = 7
Answer: 7 ✓
Unit digit = 3
Cyclicity = 4 (pattern: 3, 9, 7, 1)
47 ÷ 4 = Remainder 3
3rd position = 7
Answer: 7 ✓
Example 3: 2¹⁰⁰ ka unit digit?
Unit digit = 2
Cyclicity = 4 (pattern: 2, 4, 8, 6)
100 ÷ 4 = Remainder 0
Jab remainder 0 ho, toh last position (4th) = 6
Answer: 6 ✓
Unit digit = 2
Cyclicity = 4 (pattern: 2, 4, 8, 6)
100 ÷ 4 = Remainder 0
Jab remainder 0 ho, toh last position (4th) = 6
Answer: 6 ✓
⚠️ Important Rule:
Agar remainder = 0 aaye, toh cyclicity ki last position ka digit answer hai!
Agar remainder = 0 aaye, toh cyclicity ki last position ka digit answer hai!
Special Cases & Shortcuts
Case 1: Numbers Ending in 0, 1, 5, 6
Kisi bhi power me unit digit same rahega!
15¹⁰⁰⁰ ka unit digit = 5 ✓
26⁹⁹⁹ ka unit digit = 6 ✓
31⁵⁰ ka unit digit = 1 ✓
26⁹⁹⁹ ka unit digit = 6 ✓
31⁵⁰ ka unit digit = 1 ✓
Case 2: Even Power vs Odd Power
4 aur 9 ke liye:
4odd → unit digit = 4
4even → unit digit = 6
9odd → unit digit = 9
9even → unit digit = 1
4odd → unit digit = 4
4even → unit digit = 6
9odd → unit digit = 9
9even → unit digit = 1
Case 3: Product Ka Unit Digit
Product ka unit digit = Sabhi unit digits ka product ka unit digit
Example: 23 × 47 × 89 ka unit digit?
Unit digits: 3, 7, 9
3 × 7 = 21 (unit = 1)
1 × 9 = 9
Answer: 9 ✓
Unit digits: 3, 7, 9
3 × 7 = 21 (unit = 1)
1 × 9 = 9
Answer: 9 ✓
Case 4: Sum/Difference Ka Unit Digit
Example: 456 + 789 ka unit digit?
Unit digits: 6 + 9 = 15
Answer: 5 ✓
Unit digits: 6 + 9 = 15
Answer: 5 ✓
Power Tower (เคฎीเคจाเคฐ เคตाเคฒे Questions) TRICKY
Example: 73² ka unit digit kya hoga?
Step 1: Pehle upar wala power solve karo
3² = 9
Step 2: Ab 7⁹ ka unit digit nikalo
9 ÷ 4 = Remainder 1
1st position = 7
Answer: 7 ✓
Step 1: Pehle upar wala power solve karo
3² = 9
Step 2: Ab 7⁹ ka unit digit nikalo
9 ÷ 4 = Remainder 1
1st position = 7
Answer: 7 ✓
Example 2: 34⁵ ka unit digit?
4⁵ = 1024
Ab 3¹⁰²⁴ ka unit digit
1024 ÷ 4 = Remainder 0
Last position (4th) = 1
Answer: 1 ✓
4⁵ = 1024
Ab 3¹⁰²⁴ ka unit digit
1024 ÷ 4 = Remainder 0
Last position (4th) = 1
Answer: 1 ✓
Factorial Ka Unit Digit
n! (n ≥ 5) ka unit digit hamesha 0 hota hai
Reason: 5! = 5 × 4 × 3 × 2 × 1 = 120 (unit = 0)
6! = 720 (unit = 0)
10! = 3628800 (unit = 0)
100! ka bhi unit digit = 0 ✓
6! = 720 (unit = 0)
10! = 3628800 (unit = 0)
100! ka bhi unit digit = 0 ✓
Specific values:
1! = 1 (unit = 1)
2! = 2 (unit = 2)
3! = 6 (unit = 6)
4! = 24 (unit = 4)
5! onwards = 0 ✓
1! = 1 (unit = 1)
2! = 2 (unit = 2)
3! = 6 (unit = 6)
4! = 24 (unit = 4)
5! onwards = 0 ✓
Last 2 Digits Kaise Nikale?
Last 2 digits nikalne ke liye hame modulo 100 use karna padta hai.
Method 1: Binomial Theorem (Numbers Near 100)
Example: 97³ ka last 2 digits?
97 = (100 - 3)
97³ = (100 - 3)³
= 100³ - 3×100²×3 + 3×100×3² - 3³
Last 2 digits me sirf constant term count hoga:
= -27 (ignore negative)
= 100 - 27 = 73
Answer: 73 ✓
(Verification: 97³ = 912673)
97 = (100 - 3)
97³ = (100 - 3)³
= 100³ - 3×100²×3 + 3×100×3² - 3³
Last 2 digits me sirf constant term count hoga:
= -27 (ignore negative)
= 100 - 27 = 73
Answer: 73 ✓
(Verification: 97³ = 912673)
Method 2: Last 2 Digits of Powers
an (mod 100) nikalne ke liye:
Cyclicity check karo last 2 digits ki
Cyclicity check karo last 2 digits ki
Example: 13⁵ ka last 2 digits?
13¹ = 13
13² = 169 (last 2 = 69)
13³ = 13 × 169 = 2197 (last 2 = 97)
13⁴ = 13 × 2197 = last 2 digits of 13 × 97 = ...61
13⁵ = 13 × ...61 = last 2 of 13×61 = last 2 of 793 = 93
Answer: 93 ✓
13¹ = 13
13² = 169 (last 2 = 69)
13³ = 13 × 169 = 2197 (last 2 = 97)
13⁴ = 13 × 2197 = last 2 digits of 13 × 97 = ...61
13⁵ = 13 × ...61 = last 2 of 13×61 = last 2 of 793 = 93
Answer: 93 ✓
SSC CGL Level Practice Questions
Q1. 3⁴⁵ + 4⁴⁵ ka unit digit kya hoga?
(A) 1
(B) 3
(C) 7
(D) 9
Answer: (C) 7
Solution:
3⁴⁵: 45 ÷ 4 = R 1 → 1st position = 3
4⁴⁵: Odd power → 4
Sum = 3 + 4 = 7 ✓
(A) 1
(B) 3
(C) 7
(D) 9
Answer: (C) 7
Solution:
3⁴⁵: 45 ÷ 4 = R 1 → 1st position = 3
4⁴⁵: Odd power → 4
Sum = 3 + 4 = 7 ✓
Q2. 7¹⁰⁰ - 3¹⁰⁰ ka unit digit?
(A) 0
(B) 2
(C) 4
(D) 6
Answer: (A) 0
Solution:
7¹⁰⁰: 100 ÷ 4 = R 0 → 4th position = 1
3¹⁰⁰: 100 ÷ 4 = R 0 → 4th position = 1
Difference = 1 - 1 = 0 ✓
(A) 0
(B) 2
(C) 4
(D) 6
Answer: (A) 0
Solution:
7¹⁰⁰: 100 ÷ 4 = R 0 → 4th position = 1
3¹⁰⁰: 100 ÷ 4 = R 0 → 4th position = 1
Difference = 1 - 1 = 0 ✓
Q3. (17²³ × 23¹⁷) ka unit digit?
(A) 1
(B) 3
(C) 7
(D) 9
Answer: (D) 9
Solution:
17²³: unit digit 7, 23 ÷ 4 = R 3 → 3rd position = 3
23¹⁷: unit digit 3, 17 ÷ 4 = R 1 → 1st position = 3
Product = 3 × 3 = 9 ✓
(A) 1
(B) 3
(C) 7
(D) 9
Answer: (D) 9
Solution:
17²³: unit digit 7, 23 ÷ 4 = R 3 → 3rd position = 3
23¹⁷: unit digit 3, 17 ÷ 4 = R 1 → 1st position = 3
Product = 3 × 3 = 9 ✓
Q4. 2²⁰²⁵ ka unit digit kya hoga?
(A) 2
(B) 4
(C) 6
(D) 8
Answer: (A) 2
Solution:
2025 ÷ 4 = R 1
1st position in pattern (2,4,8,6) = 2 ✓
(A) 2
(B) 4
(C) 6
(D) 8
Answer: (A) 2
Solution:
2025 ÷ 4 = R 1
1st position in pattern (2,4,8,6) = 2 ✓
Q5. 51! ka unit digit kya hai?
(A) 0
(B) 1
(C) 5
(D) Cannot be determined
Answer: (A) 0
Solution:
n ≥ 5 ke liye n! ka unit digit = 0 ✓
(A) 0
(B) 1
(C) 5
(D) Cannot be determined
Answer: (A) 0
Solution:
n ≥ 5 ke liye n! ka unit digit = 0 ✓
Rapid Fire Exam Tricks
๐ Trick 1: Power of 2
2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32...
Pattern: 2 → 4 → 8 → 6 → 2...
Har 4th power par 6 aata hai!
2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32...
Pattern: 2 → 4 → 8 → 6 → 2...
Har 4th power par 6 aata hai!
๐ Trick 2: Consecutive Numbers Ka Product
Consecutive numbers ka product ka unit digit hamesha even (except 1×2)
Consecutive numbers ka product ka unit digit hamesha even (except 1×2)
๐ Trick 3: Even × Odd
Even number ending in 2,4,6,8 ka odd power se multiply = unit digit even
Even number ending in 2,4,6,8 ka odd power se multiply = unit digit even
Homework Practice (Khud Try Karo)
- 8⁹⁹ ka unit digit kya hoga?
- 123 × 456 × 789 ka unit digit nikalo
- 7⁵⁰ + 9⁵⁰ ka unit digit kya hoga?
- 34⁵ ka unit digit? (Power tower)
- 99³ ka last 2 digits kya honge?
Common Mistakes (Ye Galtiyan Mat Karna)
❌ Mistake 1: Pura number calculate karna
7²⁵ ko fully calculate karna time waste ✗
Cyclicity use karo ✓
7²⁵ ko fully calculate karna time waste ✗
Cyclicity use karo ✓
❌ Mistake 2: Remainder 0 ka case bhoolna
Jab remainder = 0, toh last position use karo ✓
Jab remainder = 0, toh last position use karo ✓
❌ Mistake 3: Cyclicity pattern galat yaad karna
2 ka pattern: 2,4,8,6 (not 2,4,6,8) ✓
2 ka pattern: 2,4,8,6 (not 2,4,6,8) ✓
Next Topic Preview
๐ Agle Post Me: 1.8 Factorials & Number of Zeros
Factorial calculations, trailing zeros count karna, factorial properties aur SSC level questions!
Factorial calculations, trailing zeros count karna, factorial properties aur SSC level questions!
Conclusion
Unit Digit aur Last 2 Digits ki mastery se tumhari calculation speed 10x badh jayegi! Cyclicity patterns yaad kar lo – ye exam me time saver hain.
Daily Practice: Random large powers ka unit digit mentally calculate karo. Pattern recognition improve hogi!
✅ Post Complete!
1.1-1.7 Complete ✅
Progress: 7/76 posts complete | 69 remaining
1.1-1.7 Complete ✅
Progress: 7/76 posts complete | 69 remaining