p is a positive integer. What is the remainder when 19^p is

Question

p is a positive integer. What is the remainder when 19^p is divided by 10?

1) p is a multiple of 5
2) p is a multiple of 6

banerjeea_98 · Answer

I pick "B".

state 1: p = 0...remainder is 1....p = 5...remainder....9...insuff

state 2: p= 0...remainder is 1....p= 6 ....remainder is....1....insuff

FN · Answer

B it is...

lets just focus on the unit digit of 19^p

9^0=1, 9^5=1 (i am just looking for unit digit) 

i)since p is a multiple of 5, p can be 10, 15 etc.... therefor if unit digit 1 or 9, the remainder will be diff...ins (look for 9^even or 9^odd)!

ii) since a multiple of 6 is always even, 9^even unit digit is always 1, therefore we can now the remainder for 10 for sure...sufficient!

ian7777 · Answer

all of you are right - the answer is B

greenandwise · Answer

The answer is B:

Right as explained in other posts multiples of 5 fluctuate between even and odd numbers changing the units digit from 1 to 9

Multiples of 6 will always be even making the untis digit the same (in this case 1).

GREAT QUESTION!

Antmavel · Answer

B too, very good question ...

lastochka · Answer

I don't get it. How can the people above use P=0 as an example, when it states that p is a positive integer. Now, if P=10, then the remainder should be 9, correct? And if that's true, then the answer should be, D?

Any thoughts?

gsr · Answer

B
I don't know why they use '0'. But this is how I did it now.
All even 'p' gives remainder of 1
All odd 'p' gives remainder of 9

multiples of 5 can be odd or even as in 5 or 10. So this is inconclusive.
multiples of 6 are all even. So remainder is always 1.

Titleist · Answer

lastochka

here is why when P=10 the remainder is NOT nine


P=1 2 3 4 5 6 7 8 9 10...
R=9 1 9 1 9 1 9 1 9 1 ...

The remainders change at odds and evens. The remainder is 9 when P is odd and the remainder is 1 when P is even. Thus, that is why R is 9 when P is 5 and R is 1 when P is 10 - because a multiple of 5 can be odd OR even. Since all multiples of 6 are even is why the remainder will always be 1.

Antmavel · Answer

same strategy :wink:

lastochka · Answer

appreciate the explanation. my sequence was flawed, I see the error now. thanks.

lastochka · Answer

appreciate the explanation. my sequence was flawed, I see the error now. thanks.

HongHu · Answer

Yes, pick p=0 makes absolutely no sense since the stem specified p as positive. P=0 doesn't help to pick B as the answer too. baner himself said that if p=0 then 2) would be insufficient, and then he picked B. Obviously he needed some of those study pills too, or some pushups.

p is a positive integer. What is the remainder when 19^p is

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