2008 iTest Problems/Problem 92
Problem
Find [the decimal form of] the largest prime divisor of .
Solutions
Solution 1
Using the definition of base numbers, . Let , so the number equals .
By using the Rational Root Theorem, is a factor of , so the polynomial factors into .
The first three terms share a common factor of , and the last two terms is a sum of cubes, so the expression can be grouped and factored as .
To factor the quintic polynomial, add and subtract to get . Factoring out in the first two terms results in , and factoring by grouping results in .
Thus, the polynomial can be factored into , and substituting results in . A prime test shows that is the largest prime factor of in decimal form.
Solution 2
By using a calculator or careful arithmetic, in base 10. Now go through prime factors to prime factorize .
Using the divisibility tricks, is not divisible by 2, 3, or 5. However, is divisible by 7, and factoring results in .
With careful testing, primes from 11 to 29 are not factors of . However, 31 is a factor of , so can be factored as .
Continuing on the prime hunt, we find that 37 and 41 are not factors of . However, is a factor of , so .
A prime check reveals that the number is a prime number, so the largest prime factor of is . Note that some of the factor checks can be sped up with a calculator, especially calculators that can factor numbers.
See Also
2008 iTest (Problems) | ||
Preceded by: Problem 91 |
Followed by: Problem 93 | |
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