Difference between revisions of "2000 AMC 12 Problems/Problem 6"
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<math> \mathrm{(A) \ 21 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 } </math> | <math> \mathrm{(A) \ 21 } \qquad \mathrm{(B) \ 60 } \qquad \mathrm{(C) \ 119 } \qquad \mathrm{(D) \ 180 } \qquad \mathrm{(E) \ 231 } </math> | ||
− | == Solution == | + | == Solution 1 == |
Let the primes be <math>p</math> and <math>q</math>. | Let the primes be <math>p</math> and <math>q</math>. | ||
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But <math>(p-1) \cdot (q-1)=120=2^3\cdot 3 \cdot 5</math> could be <math>2 \cdot 60,4 \cdot 30,6 \cdot 20</math> or <math>10 \cdot 12.</math> Of these, three have <math>p</math> and <math>q</math> prime, but only the last has them both small enough. Therefore the answer is <math> C </math>. | But <math>(p-1) \cdot (q-1)=120=2^3\cdot 3 \cdot 5</math> could be <math>2 \cdot 60,4 \cdot 30,6 \cdot 20</math> or <math>10 \cdot 12.</math> Of these, three have <math>p</math> and <math>q</math> prime, but only the last has them both small enough. Therefore the answer is <math> C </math>. | ||
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+ | == Solution 2 == | ||
+ | |||
+ | Since all prime numbers between 4 and 18 have an odd product and an even sum. Any odd number minus an even number is an odd number, so we can eliminate A, B, and D. We can eliminate E because the highest two prime we can pick are 13 and 17. their product is 221 and their sum is 30, so the highest number we can make is 191. Therefore, the answer is <math> C </math>. | ||
== See also == | == See also == |
Revision as of 13:29, 19 October 2010
Contents
[hide]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
Since all prime numbers between 4 and 18 have an odd product and an even sum. Any odd number minus an even number is an odd number, so we can eliminate A, B, and D. We can eliminate E because the highest two prime we can pick are 13 and 17. their product is 221 and their sum is 30, so the highest number we can make is 191. Therefore, the answer is .
See also
2000 AMC 12 (Problems • Answer Key • 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 | |
All AMC 12 Problems and Solutions |