3^50 divided by 105 leaves the remainder of?

Question

A. 3
B. 9
C. 15
D. 21
E. 27

chetan2u · Answer

Finding pattern in the given way would involve a lot of calculations. So let us work on expansion of the term.

3^{50}=3^2*(3^6)^{8}=3^2*(729)^{8}=9*(735-6)^8

As 735 is divisible by 105, the remainder will be the last term of the expansion, that is  (-6)^8=6^8

Remainder = 9*6^8=3*3*36^4=(3*36)^2*36^2=108^2*36^2=(105+3)^2*36^2 
Now  108^2  will leave  3^2  as the remainder.

Remainder =  3^2*36^2=108^2=(105+3)^2 
Again the remainder when the term is divided by 105 will be the last term  3^2  or 9

Edit: Good  Pooki , this is what I was talking of above: A 30 seconds approach. 
 This approach may not work in all such questions but where the divisor is ending with 5 or 0.

Units digit of  3^{50}=3^{4*12+2}  will be 3^2 or 9.
The units digit of 105 is 5, so the multiple of 105 will always have units digit 5 or 0.

Remainder is always positive, so abc9-abx0 or abc9-abx5 will be positive.

1) If the units digit of multiple of 105 is 0, abc9-abx0 is positive and units digit of the remainder will be 9. It could be 9, 19, 29, and so on. 
ONLY B has a units digit 9.

2) If the units digit of multiple of 105 is 5, abc9-abx5 is positive and units digit of the remainder will be 4. It could be 4, 14, 24, and so on. 
No options have a units digit 4.

B

Kinshook · Answer

Asked: 3^50 divided by 105 leaves the remainder of?

105 = 3*5*7

3^50 / 105 = 3^49 / 35

3^5 = 243 = 35*7 - 2
3^49 = 3^(5*9)*3^4
3^49 mod 35 = (-2)^9*81 mod 35 = -2^9*11 mod 35 = -32*16*11 mod 35 = 3*16*11 mod 35 = 33*16 mod 35 = -2*16 mod 35 = - 32 mod 35 = 3 mod 35 = 3

3^50 mod 105 = 3*3 = 9

IMO B

Pooki · Answer

Answer B, i use this approach which i think give you an answer so fast
3^50 will have a unit digit 9 , lets say
3^ 50= 105*x + A
We need to find A
105*x will always has a unit digit of 0 or 5
if 0 , then A must be 9
if 5 then A must be 4, look at the answer choice only 9 match

harshgoyal · Answer

Is there any simple method which can get answer in 30 seconds  Bunuel

sujoykrdatta · Answer

First, let's reduce 3 from numerator and denominator to be able to use Euler's method.

So we have: 3^49 divided by 35. Here 3 and 35 are coprime.

Euler's number of 35 
= 35(1-1/5)(1-1/7) = 24

Using Euler's method, we have: 3^24 ÷ 35 leaves remainder 1. 
So 3^48 ÷ 35 also leaves rem 1

So 3^49 ÷ 35 leaves rem 1×3 = 3

Since we had reduced the original q by 3, we should multiply it back. So the actual rem is 3×3 = 9

But this is beyond the scope of GMAT.

Alternatively 
You can see that 3^50 has units digit 9 (pattern is 3/9/7/1)
This is divided by 105.
Any multiple of 105 will have units digit 0 or 5
Hence, rem will be 9 or 4. Since 4 isn't in the options, it is 9. However this isn't a foolproof method.

If you want a method without Euler's then there are many solutions that 
 have been given above

Carcass · Answer

Great explanation. neat as always

sujoykrdatta · Answer

Thank you for the kind words!

Happy to help

3^50 divided by 105 leaves the remainder of?

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