yb wrote:

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

Explanations please!

n= 3k+2 (k is integer) and t= 5m+3 (m is integer)
1. n-2 is divisible by 5 , n-2= 3k ----> 3k is divisible by 5 -----> 3k is divisible by 15 ----> n-2 is divisible by 15

nt= (n-2)*t+2t -----> the remainder then nt is divided by 15 is determined by 2t ( becaue (n-2) is divisible by 15 ---> (n-2)*t is divisible by 15)

2t = 2*(5m+3) = 10m + 6

m=1 --> 2t=16 ---> the remainder is 1

m=0 ----> 2t= 6 ----> the remainder is 6

-----> stmt 1 is insuff

2. t is divisible by 3 ----> 5m+3 is divisible by 3 ----> 5m is divisible by 3 ----> 5m is divisible by 15 -----> 5m+3 divided by 15 has remainder of 3 ----> t divided by 15 has remainder of 3 ----> t= 15v + 3 ( v is integer)

nt= (3k+2) ( 15v+3) = 45kv + 9k +30v + 6 ----> the remainder is determined by 9k+6

k=0 ----> the remainder is 6

k= 1 ----> the remainder is 0

----> stmt 2 is insuff

Combine 1 and 2 nt= (n-2)t + 2t

(n-2)t is divisible by 15 as proved above ---> the remainder is determined by 2t

2t= 2* ( 15v+3)= 30 v +6 ----> 2t divided by 15 has remainder of 6

-----> nt is divided by 15 has remainder of 6 ---->suff

C it is.