2008 iTest Problems/Problem 92

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Problem

Find [the decimal form of] the largest prime divisor of $100111011_6$.

Solutions

Solution 1

Using the definition of base numbers, $100111011_6 = 6^8 + 6^5 + 6^4 + 6^3 + 6 + 1$. Let $x = 6$, so the number equals $x^8 + x^5 + x^4 + x^3 + x + 1$.


By using the Rational Root Theorem, $x+1$ is a factor of $x^8 + x^5 + x^4 + x^3 + x + 1$, so the polynomial factors into $(x+1)(x^7 - x^6 + x^5 + x^3 + 1)$.


The first three terms share a common factor of $x^5$, and the last two terms is a sum of cubes, so the expression can be grouped and factored as $(x+1)(x^5 (x^2 - x + 1) + (x+1)(x^2 - x + 1) = (x+1)(x^2 - x + 1)(x^5 + x + 1)$.


To factor the quintic polynomial, add and subtract $x^2$ to get $x^5 - x^2 + x^2 + x + 1$. Factoring out $x^2$ in the first two terms results in $x^2 (x^3 - 1) + x^2 + x + 1 = x^2 (x-1)(x^2 + x + 1) + x^2 + x + 1$, and factoring by grouping results in $(x^2 + x + 1)(x^3 - x^2 + 1)$.


Thus, the polynomial can be factored into $(x+1)(x^2 - x + 1)(x^2 + x + 1)(x^3 - x^2 + 1)$, and substituting $x = 6$ results in $7 \cdot 31 \cdot 43 \cdot 181$. A prime test shows that $\boxed{181}$ is the largest prime factor of $100111011_6$ in decimal form.

Solution 2

By using a calculator or careful arithmetic, $100111011_6 = 1688911$ in base 10. Now go through prime factors to prime factorize $1688911$.


Using the divisibility tricks, $1688911$ is not divisible by 2, 3, or 5. However, $1688911$ is divisible by 7, and factoring results in $7 \cdot 241273$.


With careful testing, primes from 11 to 29 are not factors of $241273$. However, 31 is a factor of $241273$, so $1688911$ can be factored as $7 \cdot 31 \cdot 7783$.


Continuing on the prime hunt, we find that 37 and 41 are not factors of $7783$. However, $43$ is a factor of $7783$, so $1688911 = 7 \cdot 31 \cdot 43 \cdot 181$.


A prime check reveals that the number $181$ is a prime number, so the largest prime factor of $100111011_6$ is $\boxed{181}$. 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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