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Zorluk: KolayRemainders and Units Digit Cyclicity

In number theory, the units digit of a positive integer raised to successive positive integer powers follows a repeating cyclic pattern. What is the units digit of 4254^{25}?

Cevap: 4

Cevap

The units digit of 4254^{25} is 4.
The units digit of powers of 4 alternates between 4 (for odd powers) and 6 (for even powers). Because 25 is an odd number, 4254^{25} has a units digit of 4.

Adım Adım Çözüm

1
Identify the units digit pattern for powers of 4
The units digits cycle between 4 (for odd exponents) and 6 (for even exponents), giving a cycle length of 2.
Units digits of positive integer powers follow a periodic pattern determined by the base digit.
2
Determine the parity of the exponent 25
25 is an odd integer (remainder 1 when divided by 2).
The exponent's remainder modulo 2 determines which position in the 2-element cycle [4, 6] the number falls into.
3
Select the corresponding units digit from the cycle
Since 25 is odd, the units digit is 4.
Odd powers of 4 always have a units digit of 4.

Anahtar Kavram

Units Digit Cyclicity
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Konular

Remainders and Units Digit Cyclicity

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