from: category_eng |
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The positive integers and are all prime numbers. The sum of these four primes is ' |
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For how many integers is the square of an integer? ' |
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Suppose that is an integer. Which of the following statements must be true about ? ' |
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7. 0.45 |
Proof |
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8. 2 |
Complex |
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9. 328,245,326 |
Application |
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10. 6641 |
Complex |
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11. 2,000,001 |
Complex |
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12. 179 | |
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Since and must have the same parity, and since there is only one even prime number, it follows that and are both odd. Thus one of is odd and the other even. Since , it follows that (as a prime greater than ) is odd. Thus , and are consecutive odd primes. At least one of is divisible by , from which it follows that and . The sum of these numbers is thus , a prime, so the answer is . |
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Since , , and are digits, , , . |
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