sebastianojara wrote:
HI! Ive been struggling with this question. Shouldnt the product of Q and P be necessary an even number? And why should 17 be the minimum value if we also have 7?
sebastianojara &
ZulfiquarA - I will take a shot to explain. Let me know if it helps?
Quote:
Shouldnt the product of Q and P be necessary an even number?
Yes, \(PQ\) will be even. The question does not state that \(PQ\) equals \(3,213\) (odd integer) rather it states that \(PQ\) is a multiple of \(3,213\)
Let's assume the \(PQ = 6\) (even). Here we can also say that \(PQ\) is a multiple of odd integer \(3\) even though \(PQ\) is even
Quote:
And why should 17 be the minimum value if we also have 7?
1. \(3,213 = 3^3 * 7 * 17\)
2. We are told that \(PQ\) is a multiple of \(3,213\) i.e. \(3,213k\) where \(k\) is a positive integer. So \(PQ = 3,213 * 2\) or \(3,213 * 4\) or \(3,213 * 6\) and so on...
3. Let's take the smallest value \(3,213 * 2\) for understanding purpose
4. \(P * Q\) \(=\) \((2 * 4 * 6 * ... * F) * (1 * 3 * 5 * ... * F)\) \(=\) Some even number
5. Now for \(PQ\) to be \(3,213 * 2\) it must also contain all prime factors and factors of \(3,213\). We need \(PQ\) to contain three \(3\)'s, one \(7\), one \(17\) and one \(2\)
6. We first need three \(3\)'s. Substitute \(F = 9\) in pt #4 and you will see that the product contains three \(3\)'s, one from \(3\) itself and other two from \(9\). You will not find the three \(3\)'s if \(F < 9\)
7. Now, we also need a \(7\) and a \(2\). Again \(F = 9\) contains a \(7\) and a \(2\) inside it
8. Next, we need a \(17\). Tell me what value of \(F\) gives a \(17\)? Only \(F >= 17\) will give you a \(17\) since \(17\) also happens to be prime number and you will not find it if \(F < 17\)
9. To summarize, \(F = 17\) is the minimum value that contains three \(3\)'s, one \(7\), one \(17\) and one \(2\) because if \(F < 17\) then \(PQ\) cannot be a multiple of \(3,213\)
Hope it helps!