Question 116193
Here's a problem: Let say you are on a construction project and on the 1st day 
you bought a certain number of 2 by 12's and paid by check, $1843. On the 2nd day you bought a different number of 2 by 12's for the same price and paid by check, $1957. As luck would have it you lost the itemized receipts. The accountant tells you I have to know the number of items, and their price for you    to get reimbursed.  You think hard and although you can't remember the price, you remember that it was a whole $ amt and a prime number:
:
Let the item cost = c;
:
Let x = no. of boards bought the 1st day; y = no. of boards on the 2nd day
:
cx = 1843 and cy = 1957, they have a common factor
:
We know that if 1843 and 1957 have a common factor, the difference will have the
same factor. Finding the difference will give a much small number to deal with
1957 - 1843 = 114; find the prime factors of 114: 2, 57,; factor 57 to 3, 19
Thus if 1843 and 1957 have a common factor it has to be 2, 3, 19, checks reveal
that it's obviously not 2, 3 doesn't work either, but what about 19?
1843/19 = 97 boards on the 1st day and 1957/19 = 103 boards on the 2nd day at $19 a board. The bean counter is happy and impressed by your math skill, and offer you a better position with firm, which started you on the road to success all because you know how to prime factor. (I got carried away)