If n is a positive integer, how many of

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NandishSS: Master | Next Rank: 500 Posts; Posts: 366; Joined: Fri Jun 05, 2015 3:35 am; Thanked: 3 times; Followed by:2 members

If n is a positive integer, how many of

by NandishSS » Wed Nov 30, 2016 9:32 pm

If n is a positive integer, how many of the ten digits from 0 through 9 could be the units digits of n^3 ?

A. Three
B. Four
C. Six
D. Nine
E. Ten

OA:E

Source:GMATPrep EP1

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MartyMurray: Legendary Member; Posts: 2131; Joined: Mon Feb 03, 2014 9:26 am; Location: https://martymurraycoaching.com/; Thanked: 955 times; Followed by:140 members; GMAT Score:800

by MartyMurray » Wed Nov 30, 2016 9:56 pm

You could see whether for each digit there is a number that when cubed has that digit in the ones place in the cube.

The key thing for doing this quickly and efficiently is realizing that only the units digit matters. So, for instance, when considering 9Â³, we don't need to figure out the actual value, only that 81 x 9 will have a units digit of 9, because 1 x 9 = 9.

0 -- 10Â³ works.
1 -- 1Â³ works.
2 -- No odd number works. 2Â³ = 8, no. 4Â³ = 64, no. All powers of 6 end in 6, no. 8 x 8 = 64 8 x 4 = 32 8 works.
3 -- No even number works. 3Â³ = 27, no. All powers of 5 end in 5, no. 7 x 7 = 49 7 x 9 = 63 7 works.
4 -- From above, 4Â³ = 64.
5 -- All powers of 5 work.
6 -- All powers of 6 work.
8 -- From above, 2Â³ = 8.
9 -- No even number works. From above 3, 5, and 7 do not work. 9 x 9 = 81 9 x 1 = 9 9 works.

Every digit could be the units digit of nÂ³.

The correct answer is E.

Marty Murray
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DavidG@VeritasPrep: Legendary Member; Posts: 2663; Joined: Wed Jan 14, 2015 8:25 am; Location: Boston, MA; Thanked: 1153 times; Followed by:128 members; GMAT Score:770

by DavidG@VeritasPrep » Thu Dec 01, 2016 6:45 am

You could also just cube all the digits from 0 to 9 and see what happens.

0^3: ends in 0
1^3: ends in 1
2^3: ends in 8
3^3: ends in 7
4^3: ends in 4
5^3: ends in 5
6^3: ends in 6
7^3: ends in 3
8^3: ends in 2
9^3: ends in 9

All 10 produced unique results! The answer is E

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pm; Location: East Bay all the way; Thanked: 625 times; Followed by:119 members; GMAT Score:780

by Matt@VeritasPrep » Thu Dec 08, 2016 8:48 pm

Useful FYI: you can get all any units digit from 0 to 9 at the end of a cube, but remember that you CANNOT get any units digit at the end of a square (and this is a property the GMAT is more likely to test!) For instance, you can say that 1447 isn't a square simply because it ends in 7, which no square can do. (No number from 0 to 9, multiplied by itself, gives a result ending in 7.)

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NandishSS: Master | Next Rank: 500 Posts; Posts: 366; Joined: Fri Jun 05, 2015 3:35 am; Thanked: 3 times; Followed by:2 members

by NandishSS » Wed Dec 14, 2016 7:14 am

Matt@VeritasPrep wrote:Useful FYI: you can get all any units digit from 0 to 9 at the end of a cube, but remember that you CANNOT get any units digit at the end of a square (and this is a property the GMAT is more likely to test!) For instance, you can say that 1447 isn't a square simply because it ends in 7, which no square can do. (No number from 0 to 9, multiplied by itself, gives a result ending in 7.)

To Add on- No square ends with unit digit 2, 3, 7, 8.

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

by Jay@ManhattanReview » Wed Dec 14, 2016 11:07 pm

NandishSS wrote:If n is a positive integer, how many of the ten digits from 0 through 9 could be the units digits of n^3 ?

A. Three
B. Four
C. Six
D. Nine
E. Ten

OA:E

Source:GMATPrep EP1

Hi Nandish,

Adding some value to the already solved question. You must know the power cycles of unit digits for the first ten digits.

Power cycles of unit digits of the first ten digits:

0^m: 0 for all the values of positive integer m
1^m: 1 for all the values of positive integer m
2^m: (2, 4, 8, 6), (2, 4, 8, 6), ... for m = 1, 2, 3, 4 => cycle of 4
3^m: (3, 9, 7, 1), ... for m = 1, 2, 3, 4 => cycle of 4
4^m: (4, 6), (4, 6), ... for m = 1, 2 => cycle of 2
5^m: 5 for all the values of m
6^m: 6 for all the values of m
7^m: (7, 9, 3, 1), ... for m = 1, 2, 3, 4 => cycle of 4
8^m: (8, 4, 2, 6), ... for m = 1, 2, 3, 4 => cycle of 4
9^m: (9, 1), (9, 1), ... for m = 1, 2, 3, 4 => cycle of 4

The blue colored digits appear on the third iteration of multiplications (n^3), we see that all the ten digits have chances of their occurrence.

OA: E

-Jay

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

by Jay@ManhattanReview » Wed Dec 14, 2016 11:10 pm

Nandish,

What could be the answer to above question had the question is modified to:

If n is a positive unit digit integer, how many of the ten digits from 0 through 9 could be the units digits of n^3 ?

A. Three
B. Four
C. Six
D. Nine
E. Ten

-Jay

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NandishSS: Master | Next Rank: 500 Posts; Posts: 366; Joined: Fri Jun 05, 2015 3:35 am; Thanked: 3 times; Followed by:2 members

by NandishSS » Fri Dec 16, 2016 4:59 am

Jay@ManhattanReview wrote:Nandish,

What could be the answer to above question had the question is modified to:

If n is a positive unit digit integer, how many of the ten digits from 0 through 9 could be the units digits of n^3 ?

A. Three
B. Four
C. Six
D. Nine
E. Ten
-Jay

_________________
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The Ans still remains E

Because 0^3: ends in 0
1^3: ends in 1
2^3: ends in 8
3^3: ends in 7
4^3: ends in 4
5^3: ends in 5
6^3: ends in 6
7^3: ends in 3
8^3: ends in 2
9^3: ends in 9

Let me know if i'm wrong!!!

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DavidG@VeritasPrep: Legendary Member; Posts: 2663; Joined: Wed Jan 14, 2015 8:25 am; Location: Boston, MA; Thanked: 1153 times; Followed by:128 members; GMAT Score:770

by DavidG@VeritasPrep » Fri Dec 16, 2016 9:30 am

NandishSS wrote:
Jay@ManhattanReview wrote:Nandish,

What could be the answer to above question had the question is modified to:

If n is a positive unit digit integer, how many of the ten digits from 0 through 9 could be the units digits of n^3 ?

A. Three
B. Four
C. Six
D. Nine
E. Ten
-Jay

_________________
Manhattan Review GMAT Prep

Locations: New York | Tokyo | Manchester | Geneva | and many more...

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The Ans still remains E

Because 0^3: ends in 0
1^3: ends in 1
2^3: ends in 8
3^3: ends in 7
4^3: ends in 4
5^3: ends in 5
6^3: ends in 6
7^3: ends in 3
8^3: ends in 2
9^3: ends in 9

Let me know if i'm wrong!!!

Re-read Jay's question stem. If n is a positive unit digit integer, how many of the ten digits from 0 through 9 could be the units digits of n^3 ? Details are everything...

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NandishSS: Master | Next Rank: 500 Posts; Posts: 366; Joined: Fri Jun 05, 2015 3:35 am; Thanked: 3 times; Followed by:2 members

by NandishSS » Sat Dec 17, 2016 7:04 am

DavidG@VeritasPrep wrote:
NandishSS wrote:
Jay@ManhattanReview wrote:Nandish,

What could be the answer to above question had the question is modified to:

If n is a positive unit digit integer, how many of the ten digits from 0 through 9 could be the units digits of n^3 ?

A. Three
B. Four
C. Six
D. Nine
E. Ten
-Jay

_________________
Manhattan Review GMAT Prep

Locations: New York | Tokyo | Manchester | Geneva | and many more...

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The Ans still remains E

Because 0^3: ends in 0
1^3: ends in 1
2^3: ends in 8
3^3: ends in 7
4^3: ends in 4
5^3: ends in 5
6^3: ends in 6
7^3: ends in 3
8^3: ends in 2
9^3: ends in 9

Let me know if i'm wrong!!!
Re-read Jay's question stem. If n is a positive unit digit integer, how many of the ten digits from 0 through 9 could be the units digits of n^3 ? Details are everything...

@David And Jay

"The devil is in the detail"
0 is neither + nor -
So the ans would be 9
Correct me if I'm wrong!!!

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[email protected]: Elite Legendary Member; Posts: 10392; Joined: Sun Jun 23, 2013 6:38 pm; Location: Palo Alto, CA; Thanked: 2867 times; Followed by:511 members; GMAT Score:800

by [email protected] » Sat Dec 17, 2016 10:23 am

Hi NandishSS,

Yes, IF the question is rewritten so that N is a positive single-digit integer, then your answer would be correct.

GMAT assassins aren't born, they're made,
Rich

Last edited by [email protected] on Sun Dec 18, 2016 9:27 pm, edited 1 time in total.

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

by Jay@ManhattanReview » Sun Dec 18, 2016 8:22 pm

DavidG@VeritasPrep wrote:
NandishSS wrote:
Jay@ManhattanReview wrote:Nandish,

What could be the answer to above question had the question is modified to:

If n is a positive unit digit integer, how many of the ten digits from 0 through 9 could be the units digits of n^3 ?

A. Three
B. Four
C. Six
D. Nine
E. Ten
-Jay

_________________
Manhattan Review GMAT Prep

Locations: New York | Tokyo | Manchester | Geneva | and many more...

Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.
The Ans still remains E

Because 0^3: ends in 0
1^3: ends in 1
2^3: ends in 8
3^3: ends in 7
4^3: ends in 4
5^3: ends in 5
6^3: ends in 6
7^3: ends in 3
8^3: ends in 2
9^3: ends in 9

Let me know if i'm wrong!!!
Re-read Jay's question stem. If n is a positive unit digit integer, how many of the ten digits from 0 through 9 could be the units digits of n^3 ? Details are everything...

Thanks David. Yes, agree with you, "Details are everything...".

-Jay

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Jeff@TargetTestPrep: GMAT Instructor; Posts: 1462; Joined: Thu Apr 09, 2015 9:34 am; Location: New York, NY; Thanked: 39 times; Followed by:22 members

by Jeff@TargetTestPrep » Wed Dec 21, 2016 9:12 am

NandishSS wrote:If n is a positive integer, how many of the ten digits from 0 through 9 could be the units digits of n^3 ?

A. Three
B. Four
C. Six
D. Nine
E. Ten

To solve, we need to raise each digit of 0 through 9 to the third power to determine how many unique units digits we can produce.

0^3 = 0

1^3 = 1

2^3 = 8

3^3 = 27 (units digit of 7)

4^3 = 64 (units digit of 4)

5^3 = 125 (units digit of 5)

Since after the base of 5 the number starts getting fairly large, we can rely on our knowledge of units digit patterns of a number raised to a power to determine the units digits of the remaining numbers.

6^3 = units digit of 6

We should recall that 6 raised to any whole number exponent will always have a units digit of 6.

7^3 = units digit of 7

We should recall that the repeating pattern for the units digits when the base of 7 is raised to an exponent is 3-9-7-1.

8^3 = units digit of 2

We should recall that the repeating pattern for the units digits when the base of 8 is raised to an exponent is 8-4-2-6.

9^3 = units digit of 9

We should recall that the pattern for the units digits when the base of 9 is raised to an exponent is 9-1.

Thus, there are 10 possible units digits for n^3 for the integers 0 through 9.

Answer:E

Jeffrey Miller
Head of GMAT Instruction
[email protected]

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