What is the remainder when integer n is divided by 3?

Question

(1) When 8n is divided by 3, the remainder is odd.
(2) When 5n is divided by 3, the remainder is odd.

vitaliyGMAT · Answer

3 has three remainders 0, 1 and 2 and only one of them is odd - 1.

We just need to solve linear congruences:

(1) 8n = 1(mod 3), 2n = 4(mod 3), n = 2(mod 3) Sufficient.

(2) 5n = 1(mod 3), 5n = 10(mod 3), n = 2(mod 3) Sufficient.

D.

manimechian · Answer

VitaliyGMAT: Can you please explain the above modular operation??

vitaliyGMAT · Answer

Hi

Linea congruencies and their systems is not the subject that can be thoroughly elucidated in just one post. Although addition, subtraction and multiplication are fairly straightforward, division requires taking into account common factor of a remainder, divisor and number itself. May be it wasn’t correct from my side to put such solution, because you don’t need linear congruencies to solve it.

Please refer to the following solution:

(1) When 8n is divided by 3, the remainder is odd.

In other words when 8n is divided by 3 the remainder is 1 (because 3 has only one odd remainder – 1).

Now: possible remainders are 0, 1 and 2.

(Remainder of 8) * (remainder of n) = 1 .

The above equation can be achieved only in case when both 8 and n have same remainders 1 or 2 (1*1 = 1 or 2*2 = 4 = 1) Because 8 has remainder 2 upon division by 3, this leaves us with only one choice for the remainder of n – 2. Sufficient.

(2) When 5n is divided by 3, the remainder is odd.

Same logic for the second option. 5 has a remainder 2, hence n should also have remainder 2.

2*2 = 4 which has remainder 1 when divided by 3. Sufficient

Hope this helps and sorry for confusion.

Cheers

Vitaliy

akshayk · Answer

S1 -> 8n = 3P + r, r = odd numbers
3 can only have 0,1,2 as remainders => r = 1.
To try, plug in numbers.
40 = 3*13 + 1
64 = 3*31 + 1... and so on.
 Sufficient.

S2 -> 5n = 3Q + r, r = odd numbers
3 can only have 0,1,2 as remainders => r = 1.
To try, plug in numbers
10 = 3*3 + 1
25 = 3*8 + 1
 Sufficient.

Therefore, Answer is D.
N = PQ + R => IMO, this question is testing the property wherein R < P.

Mo2men · Answer

What is the remainder when integer n is divided by 3?

(1) When 8n is divided by 3, the remainder is odd.

The reminder is less the divisor. The only odd reminder possible is 1

8n = 3k + 1

Possible number of 8n:

1, 4, 7, 10, 13,  16 , 19, 22, 25, 28, 31, 34, 37,  40 , 43, 46, 49, 52, 55, 58, 61,  64

Green colored number gives Integer values for n as follows:

8n = 16....n=2.......Reminder when divided by 3 is 2

8n = 40....n=5.......Reminder when divided by 3 is 2

8n = 64....n =8.......Reminder when divided by 3 is 2

Always reminder is 2

Sufficient

(2) When 5n is divided by 3, the remainder is odd.

Use same numbers as above.

1, 4, 7,  10 , 13, 16, 19, 22,  25 , 28, 31, 34, 37,  40 , 43, 46, 49, 52,  55 , 58, 61, 64

Green colored number gives Integer values for n as follows:

5n = 10....n=2.......Reminder when divided by 3 is 2

5n = 25....n=5.......Reminder when divided by 3 is 2

5n = 40....n =8.......Reminder when divided by 3 is 2

Always reminder is 2

Sufficient

Answer: D

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What is the remainder when integer n is divided by 3?

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