vikasp99 wrote:

What is the units digit of 2^39?

A) 2

(B) 4

(C) 6

(D) 8

(E) 9

To see the theory behind cyclicity, go to

LAST DIGIT OF A POWERAs

Abhishek009 notes, to find the units digit of an integer raised to a power, find the pattern the integer follows when raised to increasing powers.

Watch only the units digits.

The number of units digits in one iteration of the pattern is "cyclicity."

\(2^1 = 2\)

\(2^2 = 4\)

\(2^3 = 8\)

\(2^4 =16\)

\(2^5 = 3[2]\)

\(2^6 = 6[4]\)

\(2^7 = ...8\)

\(2^2 = ...6\)Every four powers, the units digits have a pattern of 2, 4, 8, 6

Units digit of \(2^{39}\)?

Cyclicity is 4.

Divide the exponent by that cyclicity of 4.

\(\frac{39}{4} = 9\) . . .

with remainder, \(r = 3\)

Remainder \(r\) gives you the units digit you are looking for*

r =

3? The huge number in the prompt has the same units digit as \(2^3\)

\(2^3\) has a units digit of \(8\)

Answer D

*

If r = 2, your huge number's units digit is the same as that of \(2^2\), which is 4.

If remainder r = 1, the units digit of the unknown number is the same as \(2^1\), which is 2.

If there is no remainder, r = 0, your huge number's units digit is the same as the cyclicity's number.

If this prompt asked for the units digit of \(2^{40}\), e.g., the remainder would be 0. (Exponent/Cyclicity = 40/4 = 10). If r = 0, the units digit will be the same as \(2^4\).

_________________

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