If a is the remainder when 19932021 is divided by 10, what is the val

Question

If  a  is the remainder when  1993^{2021}  is divided by  10 , what is the value of  |a - 1| + |a - 2| + |a - 3| + |a - 4| + |a - 5| ?

A.  2 
 
B.  3 
 
C.  4 
 
D.  5 
 
E.  6

nick1816 · Answer

When you divide any number by 10, remainder we get is equal to the unit digit of that number.

a is the unit digit of  1993^{2021}  or unit digit of 3^1 = 3 {since 2021=4x+1}

|a - 1| + |a - 2| + |a - 3| + |a - 4| + |a - 5|  = 2+1+0+1+2 = 6

sujoykrdatta · Answer

The cycle of powers of 3 is: 3, 9, 7, 1

The remainder when the number is divided by 10 is essentially the units digit of the number.

Thus, the units digit of 1993^2021 = units digit of 3^2021
= Units digit of 3^1 (since 2021 is 1 more than a multiple of 4)
= 3

Thus: a = 3

=> |a - 1| + |a - 2| + |a - 3| + |a - 4| + |a - 5| = 2+1+0+1+2 = 6

Answer E

Posted from my mobile device

MathRevolution · Answer

=>

3^1 = 3~3, 3^2 = 9~9, 3^3 = 27~7, 3^4 = 81~1, 3^5~3, …

So, the units digits of  3^n  have a period of  4 .

They form the cycle  3 -> 9 -> 7 -> 1.

Thus,  3^n  has the units digit of  3  if  n  has a remainder of  1  when it is divided by  4 .

The remainder when  2021  is divided by  4  is  1 , so the units digit of  1993^{2021}  is  3  and  a = 3.

|a - 1| + |a - 2| + |a - 3| + |a - 4| + |a - 5|

= |3 - 1| + |3 - 2| + |3 - 3| + |3 - 4| + |3 - 5|

= 2 + 1 + 0 + 1 + 2

= 6

Therefore, E is the correct answer.
Answer: E

BrushMyQuant · Answer

If  a  is the remainder when  1993^{2021}  is divided by 10

Theory: Remainder of a number by 10 is same as the unit's digit of the number

( Watch this Video  to Learn  How to find Remainders of Numbers by 10 )

Using Above theory Remainder of  1993^{2021}  by 10 = unit's digit of  1993^{2021}

Unit's digit of  1993^{2021}  = Unit's digit of  3^{2021}

Now to find the unit's digit of  3^{2021} , we need to find the pattern / cycle of unit's digit of power of 3 and then generalizing it.

Unit's digit of  3^1  = 3
Unit's digit of  3^2  = 9
Unit's digit of  3^3  = 7
Unit's digit of  3^4  = 1
Unit's digit of  3^5  = 3

So, unit's digit of power of 3 repeats after every  4^{th}  number.
=> We need to divided 2021 by 4 and check what is the remainder
=> 2021 divided by 4 gives 1 remainder

=>  3^{2021}  will have the same unit's digit as  3^1  = 3

=> a = Remainder of  1993^{2021}  by 10 = 3

=>  |a - 1| + |a - 2| + |a - 3| + |a - 4| + |a - 5|  =  |3 - 1| + |3 - 2| + |3 - 3| + |3 - 4| + |3 - 5| 
=  |2| + |1| + |0| + |-1| + |-2|  = 2 + 1 + 0 + 1 + 2 = 6

( Watch this video  to Learn about the  Basics of Absolute Values )

So,  Answer will be E 
Hope it helps!

MASTER How to Find Remainders with 2, 3, 5, 9, 10 and Binomial Theorem

https://www.youtube.com/watch?v=h1HTEhmUb1w

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

If a is the remainder when 19932021 is divided by 10, what is the val

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If a is the remainder when 19932021 is divided by 10, what is the val

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards