What is the units digit of 13^2003 ?

Question

What is the units digit of  13^{2003} ?

(A) 1
(B) 3
(C) 7
(D) 8
(E) 9

BrentGMATPrepNow · Accepted Answer

Look for a pattern 13^1 = 1 3 13^2 = (13)(13) = --- 9 13^3 = (13)(13^2) = (13)(---9) = ---- 7 13^4 = (13)(13^3) = (13)(---7) = ---- 1 13^5 = (13)(13^4) = (13)(---1) = ---- 3 NOTICE that we're back to where we started. 13^5 has units digit 3 , and 13^1 has units digit 3 So, at this point, our pattern of units digits keep repeating 3, 9, 7, 1, 3, 9, 7, 2, . . . We say that we have a "cycle" of 4, which means the digits repeat every 4 powers. So, we get: 13^1 = -- 3 13^2 = --- 9 13^3 = ---- 7 13^4 = ---- 1 13^5 = ---- 3 13^6 = --- 9 13^7 = ---- 7 13^8 = ---- 1 13^9 = ---- 3 13^10 = ---- 9 etc. Notice that when the exponent is a MULTIPLE of 4 (4, 8, 12, 16, ...), the units digit will be 1 Since 2000 is a MULTIPLE of 4, we know that the units digit of 13^2000 will be 1 Continuing with the pattern: 13^2001 = -- 3 13^2002 = --- 9 13^2003 = ---- 7 Answer: C Here's an article I wrote on this topic (with additional practice questions): https://www.gmatprepnow.com/articles/un ... big-powers Cheers, Brent

EgmatQuantExpert · Answer

Given

• The number  13^{2003}

To Find

• Unit digits of the number.

Approach and Working Out

• We just need to consider the last digit of the base and the last two digits of the exponent.
• It makes the number  3^{03} 
 o The unit digit of this number is 7.

Correct Answer: Option C

NandishSS · Answer

Units digit of 3 has a cycle of 4

Hence units digit of  13^{2003}  ==>  3^{2000} * 3^{3} ==>7

Hence C

Archit3110 · Answer

13^{2003}

13^2000 *13^3
cyclicty = 13 is 4 so 13
13^3 = 7
IMO C

ckavishwar · Answer

Units digit of 3 has a cycle of 4 ; 3,9,7,1

Hence units digit of 13^2003 ==> Unit digit of 13^3 = 7

Hence C

ScottTargetTestPrep · Answer

Solution:

The units digit of 13^2003 is the same as the units digit of 3^2003. Recall that the units digit pattern of powers of 3 is 3-9-7-1. Thus, when the exponent is a multiple of 4, the units digit is 1. Therefore, the units digit of 3^2004 is 1 and going backward one step in the pattern, the units digit of 3^2003 is 7.

Answer: C

Asariol · Answer

Why are we considering the last two in this case? And how do we know to?

VoltAmpere · Answer

In solving units digit problems, we just need to determine the value of the units digit of the base raise to the exponent. In this case, however, we have a large-value exponent. Since we know that the patterns for the units digit has maximum of 4 numbers, we can use only the last two digits of the exponent.

Simplified:
Units digit of 13^2003 = units digit of 3^03 = 7
CORRECT ANSWER = C

Simplified:
Units digit of 13^2003 = units digit of 3^03 = 7
 CORRECT ANSWER = C

MathRevolution · Answer

{13}^{2003} : For unit digit consider  3^{2003}

=>  (3^4)^{500} * {3^3}

=>  3^4 * 3^3 => 3^5 * 3^2

Every number repeats itself at unit digit when raised to power '5'.

=>  3 * 3^2  = 2 7

Answer C

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What is the units digit of 13^2003 ?

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What is the units digit of 13^2003 ?

Prep Toolkit

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