General number properties

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GmatTakerNo.1: Senior | Next Rank: 100 Posts; Posts: 37; Joined: Sat Apr 24, 2010 8:27 am

by GmatTakerNo.1 » Sun Apr 25, 2010 7:30 am

I have absolutely no idea how to solve this in two minutes.

Can you explain me one approach, please?

What is the units digit of 3^17Ã—17^54Ã—135^19Ã—31^21Ã—42^17?

The answer is 0.

Thank you very much.

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Source: Beat The GMAT — Problem Solving | Sun Apr 25, 2010 7:30 am

akhpad: Legendary Member; Posts: 809; Joined: Wed Mar 24, 2010 10:10 pmFollowed by: 4 members; Thanked: 50 times

by akhpad » Sun Apr 25, 2010 7:38 am

You need to know power cycle

power cycle of 2 is 2,4,8,6
power cycle of 3 is 3,9,7,1
power cycle of 5 is 5
power cycle of 7 is 7,9,3,1

I will take only UNIT digit
3^17X17^54X135^19X31^21X42^17?

= 3 * 9 * 5 * 2 = 270 (Unit digit is ZERO); it is the way to find unit digit only.

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tpr-becky: GMAT Instructor; Posts: 509; Joined: Wed Apr 21, 2010 1:08 pmFollowed by: 85 members; Location: Irvine, CA; Thanked: 199 times; GMAT Score:750

by tpr-becky » Sun Apr 25, 2010 7:40 am

The units digit of any number is obtained by multiplying the units digits - you don't have to worry about all the other digits.

The 135^19 will always end in a 5 because a 5 to an exponent always ends in a 5. the 42^17 will end in a 2 becuase the units digits of 2 to an exponent have a pattern of 2, 4, 8, 6. if you multiply 5(2) you will always end up with a units digit and it won't matter what other numbers you multiply by.

This falls under the concept of when the math gets too hard there is a pattern - find it.

Becky
Master GMAT Instructor
The Princeton Review
Irvine, CA

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GmatTakerNo.1: Senior | Next Rank: 100 Posts; Posts: 37; Joined: Sat Apr 24, 2010 8:27 am

by GmatTakerNo.1 » Sun Apr 25, 2010 7:45 am

Thank you guys.

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harshavardhanc: Legendary Member; Posts: 526; Joined: Sat Feb 21, 2009 11:47 pm; Location: India; Thanked: 68 times; GMAT Score:680

by harshavardhanc » Sun Apr 25, 2010 9:35 am

you just need a 5 and a 2 to decide the units digit, as it will be always be 0 if you find these two numbers in a product.

135 will give you a 5 and 42 will give you a two. That's it. therefore, the units digit of this messy product will be 0.

Regards,
Harsha

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