Difference between revisions of "2007 iTest Problems/Problem 43"
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She notes that two of them have exactly <math>8</math> positive divisors each. Find the common prime divisor of those two integers. | She notes that two of them have exactly <math>8</math> positive divisors each. Find the common prime divisor of those two integers. | ||
− | == Solution == | + | ==Solution== |
+ | |||
+ | Notice that each number can be written in form <math>n(n+12)(n+18) + 320</math>, where <math>n</math> is an integer from <math>700</math> to <math>799</math>. Expanding the [[polynomial]] results in <math>n^3 + 30n^2 + 216n + 320</math>, and factoring that results in <math>(n+2)(n+8)(n+20)</math>. | ||
+ | |||
+ | <br> | ||
+ | If a number has eight positive divisors, then it’s [[prime factorization]] is in the form <math>a^7</math>, <math>a^3b</math>, or <math>abc</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are primes. The most likely option is three different [[prime|prime numbers]] being multiplied together. | ||
+ | |||
+ | <br> | ||
+ | After finding the prime numbers from <math>702</math> to <math>819</math>, the values of <math>n</math> that work are <math>731</math> and <math>749</math> (after finding one value, note that both share a factor to find the other value). Thus, the prime factorizations of the two numbers are <math>733 \cdot 739 \cdot 751</math> and <math>751 \cdot 757 \cdot 769</math>, so the common prime divisor of the two is <math>\boxed{751}</math>. | ||
+ | |||
+ | ==See Also== | ||
+ | {{iTest box|year=2007|num-b=42|num-a=44}} | ||
+ | |||
+ | [[Category:Intermediate Number Theory Problems]] |
Latest revision as of 03:22, 16 June 2018
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, Answer Key) | ||
Preceded by: Problem 42 |
Followed by: Problem 44 | |
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