What is the remainder when 3^35 is divided by 5?

Question

What is the remainder when  3^{35}  is divided by 5?

A. 0
B. 1
C. 2
D. 3
E. 4

rajatchopra1994 · Answer

Explanation:
By Cyclicity
3^1=3 -- the remainder when we divide 3 by 5 is 3;
3^2=9 -- the remainder when we divide 9 by 5 is 4;
3^3=27 -- the remainder when we divide 27 by 5 is 2;
3^4=81 -- the remainder when we divide 81 by 5 is 1;
3^5=243 -- the remainder when we divide 243 by 5 is 3 AGAIN

So cyclicity is 4. 
So, 35/4= 3rd order of cyclicity of 3 .. So Remainder is 2
IMO-C

bv8562 · Answer

3^35/5 = (3^32*3^3)/5 = ((3^2)^16*27)/5 = (9^16 * 27)/5

9/5 => Remainder(R) = -1
27/5 => R = 2

Thus, Remainder = (-1)^16 * 2 =  2 (C)

CrackverbalGMAT · Answer

Number Properties is a frequently tested area on GMAT. So, you need to practice hard and make sure you learn "thinking through" questions of Number Properties before checking out any solution.

Now, lets get to some basics! There are four ways to find Remainders. 1. Using rules of Divisibility 2.Using the Remainder Theorem 3.Using the algebraic expression Dividend = divisor * Quotient + Remainder 4. Using Cyclicity for numbers expressed as powers Here we are looking at Pt.4(Using Cyclicity for numbers expressed as powers) GMAT Tip

In such questions try to reach a remainder of 1 or -1 by pairing the factors of the dividend. We have 335 being divided by 5. Now, think of the multiplication table of 5 and try to sync it up with the product of the factors here. Since 5 x 2=10 and 3*3=9, every 9 on division with 5 shall give a remainder of -1. 35 on division with 2 gives 17 as quotient and 1 as remainder. So 3^{35} = (3^{17})^2 * 3 This means we can have 17 9s(when we have 3 and3 paired) and yet after this we shall have one 3 remaining as remainder. Every 9 would create a remainder of -1. => (-1)^17 * 3 17 9's would create (-1)^17 = -1 as remainder So we have -1 * 3 =-3 as dividend in the Numerator GMAT Tip

When dividend is less than the divisor, the quotient is zero and the remainder is the dividend itself. So the - 3 would give a remainder of -3 when divided with 5. On GMAT we cannot have negative remainders GMAT Tip

To convert negative remainder to positive ,add the dividend and divisor and here , add 5with -3. So 5+(-3) =2 Hence the final answer is +2 (option c) Devmitra Sen GMAT SME

BrushMyQuant · Answer

We need to find what is the remainder when  3^{35}  is divided by 5

Theory: Remainder of a number by 5 is same as the unit's digit of the number

( Watch this Video  to Learn  How to find Remainders of Numbers by 5 )

Using Above theory , Let's find the unit's digit of  3^{35}  first.

We can do this by finding 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 35 by 4 and check what is the remainder
=> 35 divided by 4 gives 3 remainder

=>  3^{35}  will have the same unit's digit as  3^3  = 7
=> Unit's digits of  3^{35}  = 7

But remainder of  3^{35}  by 5 cannot be more than 5
=> Remainder = Remainder of 7 by 5 = 2

So,  Answer will be C 
Hope it helps!

Watch the following video to learn the Basics of Remainders

https://www.youtube.com/watch?v=P0S6j2uTaj4

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What is the remainder when 3^35 is divided by 5?

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What is the remainder when 3^35 is divided by 5?

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