GMAT Club Around The World!!! What is the remainder when 3^50 is

Question

What is the remainder when  3^{50}  is divided by 7?

A. 1
B. 2
C. 3
D. 4
E. 5

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Bunuel · Accepted Answer

Experts' Global Video Official Explanation

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ElninoEffect · Answer

3^50 can be written as (3^3)^16 * 3^2
So, (27^16 * 9) / 7
27 leaves a remainder of -1 with 7 and -1^16 = 1
1*9/7 = 2 remainder.   B

02999 · Answer

Euler of 7 = 6. so 3^6k mod 7 =0.
Left part is - 3^2 mod 7=9 mod 7=2 .

gmatophobia · Answer

Remainder( \frac{3^{50}}{7} )

Remainder( \frac{3^{48}}{7} ) * Remainder( \frac{3^{2}}{7} )

Remainder( \frac{3^{16 * 3} }{7} ) * Remainder( \frac{3^{2}}{7} )

\frac{ - 1 ^{16} }{7}  * 2

1 * 2

= 2

100mitra · Answer

3^50 = 9^25 ÷ 7

9^1=9÷7, leaves remainder 2 ………(1)
9^2=81÷7, leaves remainder 4 ………(2)
9^3=729÷7, leaves remainder 1 …….(3)
9^4=6561÷7, leaves remainder 2 …….(4)
9^5=59049÷7, leaves remainder 4 ……….(5)
9^6=531441÷7, leaves remainder 1 ……..(6)
9^7=4782969÷7, leaves remainder 2 ….(7)

Cyclic number : 3 
Thus, 25 ÷ 3 = (reminder 1 < cyclic number)
 9^25 = 9^1 when ÷ 7, reminder 2 
 Option B

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DiaPro · Answer

3^1 - 3 ————- 
3^2 - 9 ————- 9/7 , rem = 2
3^3 - 27 ————- 27/7 , rem = 6
3^4 - 81 ————- 81/7 , rem = 4 
3^5 - 243 ————- 243/7 , rem = 2 
3^6- 729 ————- 729/7 , rem = 6

So cyclical pattern 
So 3^50 corresponds to 3^2 will give rem 2 
Op B

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Khushe · Answer

The remainder when divided by 7 will be the following for 3^5, 3^6, 3^7 -

3^5 - 243/7; R = 5 
 3^6 - 729/7; R = 1 
 3^7 - 2187/7; R = 3

The negative remainders will be -2, -6 & -4 respectively which is not the same as 5, 1 & 3. Hence, the cyclicity will be 6.

MegB07 · Answer

Can someone tell if my approach is correct?

Since cyclicality of 3 is 4, I divided 40 by 4 leaving 2 as remainder. 3^2 = 9 which I divided by 7 and remainder was 2.

Bunuel · Answer

The cyclicity you refer to there determines the units digit of 3 in a positive integer power, which will not always work here. For example, if we had 3^5, then 5 gives a remainder of 1 when dividing by 4, and 3^1 divided by 7 gives a remainder of 3. However, the actual remainder when 3^5 is divided by 7 is 5.

DeepLamp · Answer

I simply used the unit digit power rule.
Only take unit digits because remainder is determined by the units digit

3^1 3
3^2 9
3^3 (2)7
3^4 (8)1
3^5 (34)3

The Cycle keeps on repeating after every 4 powers 3 returns so at power 50, the units digit will be 9 (Last number divisible by 4 is 48 plus 2). The remainder will be 2 as 9 is divided by 7

Answer Choice: B

GMAT Club Around The World!!! What is the remainder when 3^50 is

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