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Example: Let a,b,c,d be distinct prime numbers satisfying :

2a+3b+5c+7d=162
11a+7b+5c+4d=162

Given that abcd=k. Find the number of distinct values of k?
A) 0
B) 1
C) 2
D) 3
E) 4

How we go about this? We were told in school, that n variables need n equations, but we have n-2 here. A road-block? No, a call to think deeply. Just see how we can reduce variables or increase equations.

We subtract the two equations and get 9a+4b=3d => 4b=3(d-3a)

RHS is divisible by 3, so should be LHS and therefore b=3

put this in the initial equations, and we are sure the max value of a can be =7 (i leave it to u to figure out how, a hint: all prime numbers are distinct, and we have used 3, we are left with the two smallest as 2 and 5).

Back again 3a=d-4=>d=3a+4 gives us (a,d)=(5,19),(11,37).. but clealry the second set wont work, very large values. We found the set, just by using the constraint, all are distinct primes and 3 has been used.

so we have b=3,a=5 d=19, there is no further need to go as we need the no of values of k which will obviously be 1. But for the sake of completeness we can check c=2

Seems like a marathon, but no its a 3-4 minute problem, once you start doing what I want you to !

This example is credited to Sureshbala..  

Here is a link to the original post… http://mathematics-for-iit.learnhub.com/lesson/5728-prime-numbers

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