Question on Perfect Square 1

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richachampion: Legendary Member; Posts: 698; Joined: Tue Jul 21, 2015 12:12 amFollowed by: 26 members; Location: Noida, India; Thanked: 32 times; GMAT Score:740

by richachampion » Sun Oct 16, 2016 12:01 am

Which of the following numbers is a perfect square?

A. 1266
B. 1444
C. 2022
D. 4034
E. 8122

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Source: Beat The GMAT — Problem Solving | Sun Oct 16, 2016 12:01 am

richachampion: Legendary Member; Posts: 698; Joined: Tue Jul 21, 2015 12:12 amFollowed by: 26 members; Location: Noida, India; Thanked: 32 times; GMAT Score:740

by richachampion » Sun Oct 16, 2016 12:01 am

OA: B

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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 » Sun Oct 16, 2016 4:07 am

richachampion wrote:Which of the following numbers is a perfect square?

A. 1266
B. 1444
C. 2022
D. 4034
E. 8122

The units digit of a perfect square must be 0, 1, 4, 5, 6 or 9.
Eliminate C and E.

The digital root of a perfect square must be 1, 4, 7 or 9.
To compute the digital root of an integer:
Add the digits.
If the sum is less than 10, the sum is the digital root.
123 --> 1+2+3 = 6.
If the sum is greater than or equal to 10, add the digits of the sum.
The result will be the digital root.
129 --> 1+2+9 = 12 --> 1+2 = 3.

A: 1266 --> 1+2+6+6 = 15 --> 1+5 = 6.
Since the digital root is not 1, 4, 7, or 9, A is not a perfect square.
Eliminate A.

B: 1444 --> 1+4+4+4 = 13 --> 1+3 = 4.
Since the digital root is 4, hold onto B.

D: 4034 --> 4+0+3+4 = 11 --> 1+1 = 2.
Since the digital root is not 1, 4, 7, or 9, D is not a perfect square.
Eliminate D.

The correct answer is B.

Last edited by GMATGuruNY on Mon Oct 17, 2016 3:46 am, edited 1 time in total.

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Needgmat: Master | Next Rank: 500 Posts; Posts: 234; Joined: Tue May 31, 2016 1:40 am; Thanked: 3 times

by Needgmat » Sun Oct 16, 2016 5:43 am

The digital root of a perfect square must be 1, 4, 7 or 9.

Hi GMATGuruNY ,

Just a quick question.

Can you please explain that how come digital root of a perfect square must be 7?

Many thanks in advance.

Kavin

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richachampion: Legendary Member; Posts: 698; Joined: Tue Jul 21, 2015 12:12 amFollowed by: 26 members; Location: Noida, India; Thanked: 32 times; GMAT Score:740

by richachampion » Sun Oct 16, 2016 7:14 am

GMATGuruNY wrote:The units digit of a perfect square must be 1, 4, 5, 6 or 9.

Can you please share the information that what knowledge are you using in the above?

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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 » Sun Oct 16, 2016 7:48 am

Here's another question to practice with: https://www.beatthegmat.com/to-find-the- ... 76659.html

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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richachampion: Legendary Member; Posts: 698; Joined: Tue Jul 21, 2015 12:12 amFollowed by: 26 members; Location: Noida, India; Thanked: 32 times; GMAT Score:740

by richachampion » Sun Oct 16, 2016 8:04 am

Brent@GMATPrepNow wrote:Here's another question to practice with: https://www.beatthegmat.com/to-find-the- ... 76659.html

Cheers,
Brent

Thanks Sir!

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

by [email protected] » Sun Oct 16, 2016 9:57 am

Hi All,

The world of 'math' is built around a variety of patterns - and whether you have those patterns memorized or not, you can figure many of them out by simply doing some 'brute force' work and proving that they exist.

For perfect squares, consider the first 10 positive integer squares...

1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16
5^2 = 25
6^2 = 36
7^2 = 49
8^2 = 64
9^2 = 81
10^2 = 100

The units digits of those perfect squares are limited to just a few options: 0, 1, 4, 5, 6, and 9. As you work your way up the number line, the units digit 'pattern' stays the same:

11^2 = 121
12^2 = 144
13^2 = 169
Etc.

Notice how the units digit of these three values matches the first three values in the original list - it's a pattern! In that same way, you can 'work backwards' from the units digit of a PERFECT SQUARE to determine what the original integer's unit digit is...

For example, if you know you have a perfect square that ends in a 4, then the square root of that number has a units digit that is either a 2 or an 8 (look at the original list and you'll see why). That logic/knowledge, combined with some additional brute-force arithmetic, can get you to the correct answer here fairly efficiently.

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Rich

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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 » Mon Oct 17, 2016 3:48 am

Needgmat wrote:
The digital root of a perfect square must be 1, 4, 7 or 9.
Hi GMATGuruNY ,

Just a quick question.

Can you please explain that how come digital root of a perfect square must be 7?

Many thanks in advance.

Kavin

Information about digital roots can be found here:
https://en.wikipedia.org/wiki/Digital_root

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Mo2men: Legendary Member; Posts: 712; Joined: Fri Sep 25, 2015 4:39 amFollowed by: 5 members; Thanked: 14 times

by Mo2men » Mon Oct 17, 2016 8:37 am

GMATGuruNY wrote:
richachampion wrote:Which of the following numbers is a perfect square?

A. 1266
B. 1444
C. 2022
D. 4034
E. 8122
The units digit of a perfect square must be 0, 1, 4, 5, 6 or 9.
Eliminate C and E.

The digital root of a perfect square must be 1, 4, 7 or 9.
To compute the digital root of an integer:
Add the digits.
If the sum is less than 10, the sum is the digital root.
123 --> 1+2+3 = 6.
If the sum is greater than or equal to 10, add the digits of the sum.
The result will be the digital root.
129 --> 1+2+9 = 12 --> 1+2 = 3.

A: 1266 --> 1+2+6+6 = 15 --> 1+5 = 6.
Since the digital root is not 1, 4, 7, or 9, A is not a perfect square.
Eliminate A.

B: 1444 --> 1+4+4+4 = 13 --> 1+3 = 4.
Since the digital root is 4, hold onto B.

D: 4034 --> 4+0+3+4 = 11 --> 1+1 = 2.
Since the digital root is not 1, 4, 7, or 9, D is not a perfect square.
Eliminate D.

The correct answer is B.

Hi GMATGuru,

What would be the case if two choices have the same digital root. In the following question:

Which of the following is a perfect square?

A) 649
B) 961 ............9+6+1=16....1+6.......Digital root 7
C) 1664
D) 2509............9+6+1=16....1+6.......Digital root 7
E) 100000

Thanks for your support

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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 » Mon Oct 17, 2016 8:55 am

Mo2men wrote:Hi GMATGuru,

What would be the case if two choices have the same digital root. In the following question:

Which of the following is a perfect square?

A) 649
B) 961 ............9+6+1=16....1+6.......Digital root 7
C) 1664
D) 2509............9+6+1=16....1+6.......Digital root 7
E) 100000

Thanks for your support

Check whether 961 is a perfect square.
For xÂ² to yield a units digit of 1, x must have a units digit of 1 or 9.
Since 30Â² = 900, test 31Â²:
31Â² = (30+1)Â² = 30Â² + 2*30*1 + 1Â² = 900 + 60 + 1 = 961.
Success!

The correct answer is B.

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