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Zorluk: KolayRemainders and Units Digit Cyclicity

What is the units digit of 7437^{43}?

Cevap: 3

Cevap

The units digit of 7437^{43} is 3.
The units digits of powers of 7 repeat in a pattern of four terms: 7, 9, 3, 1. Dividing the exponent 43 by 4 yields a remainder of 3. The 3rd term in the repeating pattern is 3, so the units digit of 7437^{43} is 3.

Adım Adım Çözüm

1
Determine the cyclicity pattern of the units digit of powers of 7.
The units digits for 71,72,73,74,…7^1, 7^2, 7^3, 7^4, \dots follow the repeating sequence 7, 9, 3, 1 with a cycle length of 4.
Units digits of positive integer powers follow a periodic pattern.
2
Divide the exponent 43 by the pattern cycle length of 4.
43÷4=1043 \div 4 = 10 remainder 3.
The remainder determines which term in the repeating sequence gives the units digit.
3
Find the units digit corresponding to the 3rd term in the cyclicity sequence.
The 3rd digit in the sequence (7, 9, 3, 1) is 3.
A remainder of 3 corresponds to the 3rd power in the cycle, 737^3.

Anahtar Kavram

Units Digit Cyclicity
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Konular

Remainders and Units Digit Cyclicity

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