Remainder question

[susan.meng]

When positive integer n is divided by 3, the remainder is 2 and when positive integer t is divided by 5 the remainder is 3. What is the remainder when the product nt is divided by 15?

1. n-2 is divisible by 5
2. t is divisible by 3

Known from Q stem: n = 3k + 2; t = 5p + 3
Is this rephrasing correct? nt = 15kp + 9k + 10p + 6 --> what is 9k +10p +6?

1. n-2 = 5x --> 5x+2 = 3k+2 --> x = 3 and k = 5
what's the takeaway?

2. t=3a --> 3a = 5p+3

Could you please solve using theory?

[mxs2009]

hey Susan, you've basically done the heavy lifting and now you just need to put the puzzle pieces together.

Statement 1 tells us that:
(3k+2)-2 = an integer. (3k)/5. Thus, k must be a multiple of 5
5

Statement 2 tells us that:
5p+3 = an integer or (5/3)p +1 is an integer. Thus, p must be a multiple of 3
3

Going back to your original rephrase:
nt = 15kp + 9k + 10p + 6

rewrite as, what is the remainder of:
(15kp) + (3^2*k) + (2*5*p) + (6)
15

break down the four parts:
15kp/15 has no remainder
(3^2)*k/15 has no remainder because k is a multiple of 5
2*5*p/15 has no remainder because p is a multiple of 3
6 will always be the remainder and your answer is C

Neither A nor B is sufficient alone because we don't know if the other parts divide evenly.

hope that helps

[susan.meng]

fantastic, thank you

[Ben Ku]

Thanks mxs2009, that's a nice way to use the theory of this problem. If you're more comfortable with the algebra, this way approach is extremely helpful. In this kind of problem, the plugging in numbers approach might be cumbersome: it's hard to find the right numbers to plug in, and you may need to try different possibilities. However, the advantage of plugging in numbers is that you can spot patterns.

[pramil25.pm]

I insufficient
n/3 rem 2 2,5,8,11,14,17,20
II insufficient
n/5 rem 3 3,8,13,18,23

1) N=2,7,12,17............
2) 2)t=3,6,9,12,15,18.............

1 n 2 combined

N 2,17,32..
T 3,18....

Nt 6,51,36,96 divide them by 15 the remainder is alwaz 6
So C

[jlucero]

As long as you are quick with your computations, this is a perfectly acceptable way of doing this problem. Just make sure you could solve it this way in two minutes or less.