SarwaSamadhan

Unit digit of big powers using cyclicity

बड़ी घातों का इकाई अंक — चक्रीयता (cyclicity) से

The last digit of powers repeats in short cycles: 2, 3, 7 and 8 repeat every 4; 4 and 9 every 2; 0, 1, 5 and 6 never change. Divide the power by the cycle length and use the remainder.

घातों का अंतिम अंक छोटे चक्रों में दोहराता है: 2, 3, 7, 8 हर 4 पर; 4 और 9 हर 2 पर; 0, 1, 5, 6 हमेशा वही रहते हैं। घात को चक्र की लंबाई से भाग दें और शेषफल का उपयोग करें।

The cycles

2 → 2, 4, 8, 6 · 3 → 3, 9, 7, 1 · 7 → 7, 9, 3, 1 · 8 → 8, 4, 2, 6 · 4 → 4, 6 · 9 → 9, 1 · 0, 1, 5, 6 → always themselves.

  1. Look only at the last digit of the base.
  2. Divide the power by the cycle length (4 or 2) and find the remainder.
  3. Remainder r → take the r-th item of the cycle; remainder 0 → take the last item.
Example Unit digit of 7^2023
2023 ÷ 4 leaves 3 → third item of 7, 9, 3, 1 → 3
Example Unit digit of 2^50
50 ÷ 4 leaves 2 → second item of 2, 4, 8, 6 → 4
Example Unit digit of 9^20
Even power of 9 → 1

Try it yourself · अभ्यास

  1. Unit digit of 3^45
    Show answer

    45 ÷ 4 leaves 1 → 3

  2. Unit digit of 8^17
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    17 ÷ 4 leaves 1 → 8

  3. Unit digit of 4^99
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    Odd power of 4 → 4