2000 AMC 12 Problems/Problem 6
From AoPSWiki
Contents |
Problem
Two different prime numbers between 
and 
are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
Solution 1
Let the primes be 
and 
.
The problem asks us for possible values of 
where 
Using Simon's Favorite Factoring Trick:
Possible values of 
and 
are:
The possible values for 
(formed by multipling two distinct values for and ) are:
So the possible values of are:
The only answer choice on this list is 
Note: once we apply the factoring trick we see that, since 
and 
are even, should be a multiple of .
These means that only and 
are possible.
We can't have 
with and below . Indeed, 
would have to be 
or 
.
But 
could be 
or 
Of these, three have and prime, but only the last has them both small enough. Therefore the answer is 
.
Solution 2
First, since all primes in between 4 and 18 are odd, the product is odd. This eliminates options B and D. So 
, but that's not possible. 
, which works perfectly so the answer is just 
.
- Note: this problem did not require the strategy used in Solution 1. It takes up to much of your needed time during the test. In another case, it will probably be more usefull.
See also
| 2000 AMC 12 (Problems • Resources) | ||
| Preceded by Problem 5 | Followed by Problem 7 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
