Units digit for 12^13^14^15

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gmatdriller: Master | Next Rank: 500 Posts; Posts: 418; Joined: Sun Jul 04, 2010 12:48 pmFollowed by: 3 members; Thanked: 6 times

by gmatdriller » Fri Jul 24, 2015 3:09 am

What is the units digit for the entire expression?

12^13^14^15

Thanks as usual.

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Source: Beat The GMAT — Problem Solving | Fri Jul 24, 2015 3:09 am

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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 » Fri Jul 24, 2015 6:23 am

gmatdriller wrote:What is the units digit for the entire expression?

12^13^14^15

This is a pretty time consuming question. In fact, I'd say that it requires a little too much "brute force" to be an official GMAT question.

We have two free videos on finding the units digits of large powers (https://www.gmatprepnow.com/module/gmat- ... ts?id=1031 and https://www.gmatprepnow.com/module/gmat- ... ts?id=1032) so I will defer partly to those videos.

12^13: The units digit pattern is 2, 4, 8, 6, 2, ... with cycle 4 (see videos for technique). This means the units digit of 12^12 is 6, which means the units digit of 12^13 is 2

So, 12^13^14^15 = (---2)^14^15

(---2)^14: The units digit pattern is 2, 4, 8, 6, 2, ... with cycle 4. This means the units digit of (---2)^12 is 6, which means the units digit of (---2)^14 is 4

So, 12^13^14^15 = (---2)^14^15 = (---4)^15

(---4)^15
The units digit pattern is 4, 6, 4, 6, 4,... with cycle 2. This means the units digit of (---4)^15 is 4

Answer: 4

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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gmatdriller: Master | Next Rank: 500 Posts; Posts: 418; Joined: Sun Jul 04, 2010 12:48 pmFollowed by: 3 members; Thanked: 6 times

by gmatdriller » Fri Jul 24, 2015 8:41 am

Thanks Brent.
I did it this way too but realized it may not be correct.

For example: 3^2^5 has 3 as unit digit.

Using your method;
2^5 has unit digit 2
So, 3^2 has unit digit 9

Am I missing something?

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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 » Fri Jul 24, 2015 8:51 am

gmatdriller wrote:Thanks Brent.
I did it this way too but realized it may not be correct.

For example: 3^2^5 has 3 as unit digit.

Using your method;
2^5 has unit digit 2
So, 3^2 has unit digit 9

Am I missing something?

The question could have been posed to be less ambiguous.
For example 2^5^3 can be interpreted in two ways: (2^5)^3 and 2^(5^3)
The first is equivalent to 2^15 and the second is equivalent to 2^125
I interpreted 12^13^14^15 as [(12^13)^14]^15

I believe you are using the second interpretation.

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Fri Jul 24, 2015 9:17 am

Alternate interpretation:

What is the units digit of 12^[13^(14Â¹â�µ)]?

Determine the CYCLE of units digits when 12 is raised to increasing powers:
12Â¹ --> units digit of 2
12Â² --> units digit of 4
12Â³ --> units digit of 8
12â�´ --> units digit of 6
12â�µ --> units digit of 2
12â�¶ --> units digit of 4.

The results above indicate that the units digit repeat in a CYCLE OF 4:
2, 4, 8, 6...2, 4, 8, 6...2, 4, 8, 6...
Implication:
Every exponent that is a MULTIPLE OF 4 completes a cycle, yielding a UNITS DIGIT OF 6.
From there, the cycle REPEATS:
2, 4, 8...
Thus, every exponent that is ONE MORE THAN A MULTIPLE OF 4 begins a new cycle, yielding a UNITS DIGIT OF 2.

When 13 is raised to a power, the result = (multiple of 4) + 1.
13Â¹ = 13 = 12 + 1 = (4*1) + 1.
13Â² = 169 = 168 + 1 = (4*42) + 1.
13Â³ = 2197 = 2196 + 1 = (4*549) + 1.

Thus:
12^(13^power) = 12^(4a + 1), where a is a nonzero integer.
Since the exponent is one more than a multiple of 4, the units digit will be 2.

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nikhilgmat31: Legendary Member; Posts: 518; Joined: Tue May 12, 2015 8:25 pm; Thanked: 10 times

by nikhilgmat31 » Wed Jul 29, 2015 2:18 am

Mitch- I think you missed some powers in between.

12^ 13 we can take it as 2^13
2^n= 2,4,8,6.........

for n = 13 (with groups of 4) unit digit will be 2

next step for 2^14
2^n= 2,4,8,6.........
for n=14 (with groups of 4) unit digit will be 4

next step for 5^15
4^n= 4,6,4,6,4,6
for n=15 (with groups of 2) unit digit will be 4

Answer is 4

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