2007 iTest Problems/Problem 43
Problem
Bored of working on her computational linguistics thesis, Erin enters some three-digit integers into a spreadsheet, then manipulates the cells a bit until her spreadsheet calculates each of the following one hundred -digit integers:
She notes that two of them have exactly positive divisors each. Find the common prime divisor of those two integers.
Solution
Notice that each number can be written in form , where is an integer from to . Expanding the polynomial results in , and factoring that results in .
If a number has eight positive divisors, then it’s prime factorization is in the form , , or , where , , and are primes. The most likely option is three different prime numbers being multiplied together.
After finding the prime numbers from to , the values of that work are and (after finding one value, note that both share a factor to find the other value). Thus, the prime factorizations of the two numbers are and , so the common prime divisor of the two is .
See Also
2007 iTest (Problems) | ||
Preceded by: Problem 42 |
Followed by: Problem 44 | |
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