What is remainder when 447^386 is divided by 5?

Question

What is the remainder when  447^{386}  is divided by 5?

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

globaldesi · Answer

447^386 remainder will be same as 7^386 is divided by 5
since it could be written as (440+7)^386
and first art is always divisible by 5
now
7/ 5 = Rem(2)
7^2/5 = 49/5= Rem(4)
7^3/5= Rem(3)
7^4/5 = Rem(1)
7^5/5 = Rem(2)

so it follows the cycle of 4 
384 divided by 4 leaves remainder 2 thus remainder would be 4

IanStewart · Answer

If you know the units digit of a number, you can find its remainder when you divide by 5. For example, the remainder you get when you divide 394 by 5 is just 4, and the remainder when you divide 407 by 5 is just 2.

So all we want here is the units digit of 447^386. If that digit is between 0 and 4, that will be the answer, and if it is between 5 and 9, we'll subtract 5 from it to find the answer. The units digit of 447^386 is the same as the units digit of 7^386. There are several ways to find that units digit. For example, 7^386 = (7^2)^193 = 49^193. The units digit of that number is the same as the units digit of 9^183. If you look at small powers of 9, you'll see that when you raise 9 to an odd exponent, the units digit is 9 (and when raised to an even exponent, the units digit is 1). So 9^183 has a units digit of 9, and so does 447^386, and the remainder we want is 4.

The numbers in this question are too inelegant to be realistic GMAT numbers, incidentally. The GMAT does test the same concepts, but when it does, it would use a simpler base and a simpler exponent.

Target4bschool · Answer

Last digit of multiples of 7 are in cycle: 7, 9, 3, 1, 7, 9, 3, 1, 7, ....
So, let's find the last digit of this number.
386/4 = 96 + 2 [remainder]

So, last digit will be 2nd digit, i.e. 9
9/5 will give 4 as a remainder

IMO, E

If you have further que, please ask.
Please give kudos if it helped

Fdambro294 · Answer

Concept: Divisibility and Remainder Rule for 5:

We can Find the Remainder when Dividing by 5 by looking at the Units Digit

(1st) if the Units Digit = 0 or 5 -------> Rem = 0 and Number is Divisible by 5

(2nd) If the Units Digit = 1, 2 , 3 or 4 -----> Rem = Units Digit

(3rd) If the Units Digit = 6 , 7 , 8 , or 9 ---> Rem = (Units Digit) - 5

Rule: Given the Base 447, we know that Consecutive Integer Powers will follow the Units Digit Pattern of:

(7)^X ------   ----- in which the Pattern Repeats after Every Consecutive Integers

For Example:

447^1 ------- Units Digit = 7

447^2 ------- Units Digit = 9

447^3 ------- Units Digit = 3

447^4 -------- Units Digit = 1

Step 1: Find the Units Digit of the Numerator:

(447)^386

386 / (Cyclicity of 4) = 96 + Remainder of 2

Thus, after Cycling through the Units Digit Pattern of (7)^X a Total of 96 Times, the Units Digit will be:

9

Step 2: Remainder when we Divide by 5

(447)^386 / 5 = (........4)/5 -----> which will yield a Remainder = 4

-E-

GCMEMBER · Answer

447^386 will have the units digit same as 7^386
7^386 will have units digit same as 7^2
7^2 when divided by 5 leaves a remainder 4.
Therefore, 447^386 when divided by 5 will leave a remainder 4

Posted from my mobile device

MathRevolution · Answer

447^{386} = 7^{4 * 96} * 7^2

=>  7^4 * 7*2 = 7^5 * 7

=> 7 * 7 = 49

49 when divided by '5' will give remainder as '4'.

Answer E

CrackverbalGMAT · Answer

Every 447 divided by 5 gives a remainder of 2

The question is now to find  R  . We try to make the numerator in the form (5 + 1) or (5 - 1)

R  = R  = R  = R  = R  = -1

A remainder of -1 is the same as a remainder of 5 - 1 = 4

Option E

Arun Kumar

BrushMyQuant · Answer

We know to find what is the remainder when   447^{386}   is divided by 5

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

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

Remainder of    447^{386}   will be same as remainder of  7^{386}

We can do this by finding the pattern / cycle of unit's digit of power of 7 and then generalizing it.

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

So, unit's digit of power of 7 repeats after every  4^{th}  number.
=> Remainder of 386 by 4 = 2
=> Units' digit of   447^ {386}   = Units' digit of  7^{2}  = 9

=> Remainder = Remainder of 9 by 5 = 4

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

What is remainder when 447^386 is divided by 5?

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

What is remainder when 447^386 is divided by 5?

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