17^27 has a units digit of:

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BTGmoderatorDC: Moderator; Posts: 7187; Joined: Thu Sep 07, 2017 4:43 pmFollowed by: 23 members

by BTGmoderatorDC » Wed Jan 09, 2019 8:17 pm

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17^27 has a units digit of:

(a) 1
(b) 2
(c) 3
(d) 7
(e) 9

OA C

Source: Manhattan Prep

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Source: Beat The GMAT — Problem Solving | Wed Jan 09, 2019 8:17 pm

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Jay@ManhattanReview: GMAT Instructor; Posts: 3008; Joined: Mon Aug 22, 2016 6:19 amFollowed by: 34 members; Location: Grand Central / New York; Thanked: 470 times

by Jay@ManhattanReview » Wed Jan 09, 2019 10:04 pm

BTGmoderatorDC wrote:17^27 has a units digit of:

(a) 1
(b) 2
(c) 3
(d) 7
(e) 9

OA C

Source: Manhattan Prep

Let's understand the power cycle of 7.

"¢ 7^1 = 7; unit digit = 7;
"¢ 7^2 = 49; unit digit = 9;
"¢ 7^3 = 343; unit digit = 3;
"¢ 7^4 = ...1; unit digit = 1;

"¢ 7^5 = ...7; unit digit = 7;

We see that units digit of the power of 7 repeats after every 4 cycle.

Let's see 17^27 now.

17^27 = 17^(24 + 3) = 17^(6*4 + 3)

Note that units digit of 17^27 will be the same that of 17^3 since 6*4 has a cycle of 4.

Again, the units digit of 17^3 will have the same units digit as has 7^3.

Now 7^3 = 343; thus, the units digit of 17^27 = 3.

The correct answer: C

Hope this helps!

-Jay
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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Thu Jan 10, 2019 6:38 am

BTGmoderatorDC wrote:17^27 has a units digit of:

(a) 1
(b) 2
(c) 3
(d) 7
(e) 9

Look for a pattern

17^1 = 17
17^2 = (17)(17) = ---9 [aside: we need not determine the other digits. All we care about is the units digit]
17^3 = (17)(17^2) = (17)(---9) = ----3
17^4 = (17)(17^3) = (17)(---3) = ----1
17^5 = (17)(17^4) = (17)(---1) = ----7

NOTICE that we're back to where we started.
17^5 has units digit 7, and 17^1 has units digit 7
So, at this point, our pattern of units digits keep repeating 7, 9, 3, 1, 7, 9, 3, 1, . . .
We say that we have a "cycle" of 4, which means the digits repeat every 4 powers.

So, we get:
17^1 = --7
17^2 = ---9
17^3 = ----3
17^4 = ----1
17^5 = ----7
17^6 = ---9
17^7 = ----3
17^8 = ----1
17^9 = ----7
17^10 = ----9
etc.

Notice that when the exponent is a MULTIPLE of 4 (4, 8, 12, 16, ...), the units digit will be 1
Since 24 is a MULTIPLE of 4, we know that the units digit of 17^24 will be 1
Continuing on, we get:
17^25 = ----7
17^26 = ---9
17^27 = ----3

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

Brent Hanneson - Creator of GMATPrepNow.com

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fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

by fskilnik@GMATH » Thu Jan 10, 2019 8:59 am

BTGmoderatorDC wrote:17^27 has a units digit of:

(a) 1
(b) 2
(c) 3
(d) 7
(e) 9
Source: Manhattan Prep

$$? = \left\langle {{{17}^{27}}} \right\rangle = \left\langle {{7^{27}}} \right\rangle $$
$$\left\langle {{7^4}} \right\rangle = \left\langle {{7^2} \cdot {7^2}} \right\rangle = \left\langle {\left\langle {{7^2}} \right\rangle \cdot \left\langle {{7^2}} \right\rangle } \right\rangle = 1$$
$$\left\langle {{7^{24}}} \right\rangle = \left\langle {{7^4} \cdot {7^4} \cdot \ldots \cdot {7^4}} \right\rangle = {\left\langle {{7^4}} \right\rangle ^6} = 1$$
$$? = \left\langle {{7^{24}} \cdot {7^3}} \right\rangle = \left\langle {{7^{24}}} \right\rangle \cdot \left\langle {{7^3}} \right\rangle = 1 \cdot 3 = 3$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8134; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Mon Jan 21, 2019 5:42 pm

BTGmoderatorDC wrote:17^27 has a units digit of:

(a) 1
(b) 2
(c) 3
(d) 7
(e) 9

Since we only care about units digits, we can rewrite the expression as:

7^27

Now we can evaluate the pattern of the units digits of 7^n for positive integer values of n. That is, let's look at the pattern of the units digits of powers of 7. When writing out the pattern, notice that we are ONLY concerned with the units digit of 7 raised to each power.

7^1 = 7

7^2 = 9

7^3 = 3

7^4 = 1

7^5 = 7

The pattern of the units digit of powers of 7 repeats every 4 exponents. The pattern is 7-9-3-1. In this pattern, all positive exponents that are multiples of 4 will produce a 1 as its units digit. Thus:

7^28 has a units digit of 1, and 7^27 has a units digit of 3.

Answer: C

Scott Woodbury-Stewart
Founder and CEO
[email protected]

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