12. Unit Digit & Cyclicity (Last Digit of Powers)

Unit digit (last digit) aur cyclicity ka complete concept: powers ke last digit patterns, cycle length, exponent reduction, last 2 digits (mod 100) basics, SSC CGL level examples & practice with answers.

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1) Unit digit ka meaning (simple)

Unit digit = last digit. Kisi bhi power ka unit digit nikalne ke liye sirf base ka last digit enough hota hai.

• Example: 137^k ka unit digit same as 7^k
• Example: 52^k ka unit digit same as 2^k

2) Cyclicity kya hoti hai?

Powers me last digit repeat hota hai. Is repeating pattern ko cycle bolte hain. Cycle ki length = kitne steps ke baad pattern repeat ho jaata hai.

3) Unit digit cycles (Most Important Table)

4) Standard steps (SSC method)

1. Base ka last digit lo.
2. Us last digit ka cycle length L dekho.
3. Exponent e ka e mod L nikalo.
4. Remainder 0 aaye to position = L (last term of cycle).
5. Cycle ke us position ka digit = answer.

5) Examples (clear)

Example 1: 7²⁰²¹ ka unit digit

• 7 ka cycle: 7, 9, 3, 1 (L=4)
• 2021 mod 4 = 1
• Position 1 ⇒ unit digit = 7

Example 2: 2¹⁰⁰ ka unit digit

• 2 ka cycle: 2, 4, 8, 6 (L=4)
• 100 mod 4 = 0 ⇒ position = 4
• Position 4 ⇒ unit digit = 6

Example 3: 123⁴⁵ ka unit digit

• Base last digit = 3
• 3 cycle: 3, 9, 7, 1 (L=4)
• 45 mod 4 = 1 ⇒ unit digit = 3

Example 4: (12³⁵ × 7²²) ka unit digit

• 12 last digit 2 ⇒ 2³⁵ unit digit: 35 mod 4 = 3 ⇒ 8
• 7²² unit digit: 22 mod 4 = 2 ⇒ 9
• Product unit digit: 8×9 = 72 ⇒ unit digit = 2

6) Last 2 digits (basic idea)

Last 2 digits = number mod 100. SSC me last-2-digits questions me usually: base ko 100 se reduce karte hain aur pattern (cycle) build karte hain.

Example (simple): last 2 digits of 7⁴

• 7⁴=2401 ⇒ last 2 digits = 01

Note: Complex last-2-digits powers (large exponents) me modulo rules + cyclicity combine hota hai. Agar aap chaho to next page me “last 2 digits advanced” add kar denge.

7) Common traps

• Exponent mod L me remainder 0 aaye to position L hoti hai (1 nahi).
• Base ka last digit hi use karo (poora base nahi).
• Product/power expression me stepwise unit digit nikaal ke multiply karo.

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8) Practice (SSC CGL) + Answers

1. Find unit digit of 3⁵⁷.
2. Find unit digit of 8¹²³.
3. Find unit digit of 9²⁰²⁶.
4. Find unit digit of 17⁸⁹ + 6⁵⁰.
5. Find unit digit of (5⁹⁹ × 4⁵¹).
6. Find unit digit of (2³⁵ − 7¹⁹).

Show Answers

1. 3 cycle (3,9,7,1), 57 mod 4 = 1 ⇒ 3
2. 8 cycle (8,4,2,6), 123 mod 4 = 3 ⇒ 2
3. 9 cycle (9,1), 2026 mod 2 = 0 ⇒ 1
4. 17 last digit 7 ⇒ 7⁸⁹: 89 mod 4 = 1 ⇒ 7; 6⁵⁰ unit digit always 6; 7+6=13 ⇒ unit digit 3
5. 5⁹⁹ unit digit 5; 4 cycle (4,6), 51 mod 2 = 1 ⇒ 4; 5×4=20 ⇒ 0
6. 2³⁵: 35 mod 4 = 3 ⇒ 8; 7¹⁹: 19 mod 4 = 3 ⇒ 3; 8−3=5 ⇒ unit digit 5