zaarathelab wrote:When positive integer n is divided by 3, the remainder is 2; when positive integer t is divided by 5, the remainder is 3. What is the remainder when the product is nt is divided by 15?
a) n-2 is divisible by 5
b) t is divisible by 3
I tried to do it algebraically, but I took a lot of time. What is the quickest way to do this (under 2 minutes)?
Are there any tips to know when to follow the algebraic method and when to follow the plugging in number method for divisibility/remainder problems?
Okay, so I think your solution is fine. Generally, if you're trying to prove sufficiency, an algebraic approach has some benefits, because you can never really PROVE sufficiency by showing that it works out for a few cases. However, we may be able to combine the approaches to make the purely algebraic approach less tedious.
From the given: n=3a+2, t=5b+2: n=2,5,8,11,14,17... t=3,8,13,18,23,28...
Statement 1: n-2 is divisible by 5, so: n=5c+2, n=2,7,12,17,... Combining this with the list for n above, we can see that n=15d+2. Once you find the first number that satisfies both conditions, each successive number will be LCM(5,3)=15 away, hence n=15d+2. Possible values of nt: n=2, t=3, nt=6 (remainder of 6); n=2, t=8, nt=16 (remainder of 1). INSUFFICIENT.
Statement 2: t is divisible by 3, so t=3f: t=3,6,9,12,... Combining with the list above, 3 is the smallest value that satisfies both conditions, so t=15g+3, so t=3,18,33... Possible values of nt: n=2,t=3 nt=6 (remainder of 6); n=5,t=3 nt=15 (remainder of zero) INSUFFICIENT.
Statements 1&2: n=15d+2 and t=15g+3. We can list numbers here if we want, but again, that can only make a strong case for sufficiency but it can't prove sufficiency, so it's risky. Algebraically, n*t=(15d+2)(15g+3) = 15^2*dg + 3*15d + 2*15d + 6. The first three terms are clearly divisible by 15, so the remainder of n*t when divided by 15 must always be 6. SUFFICIENT.
Whether this approach is preferable to a purely algebraic approach will depend on how comfortable you are manipulating the equations and interpreting the results quickly and accurately. If you're not that comfortable, listing some numbers can make you more confident that your equations make sense.
