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Number Theory Level 3

Find the largest number of the form $\overline {AABB}$ which is the product of three distinct prime numbers and both the numbers of the form $\overline {AAB}$ and $\overline {ABB}$ are primes

The answer is 9911.

1 solution

Joshua Lowrance
May 2, 2019

We want $AABB$ to be as big as possible, so we will start testing values that make $AABB$ as large as possible. The three requirements are: $AAB$ is prime, $AAB$ is prime, $AABB$ is product of three primes. (If one of the requirements is not met, I will skip the rest, because it won't matter). We can immediately skip all of the scenarios where $B$ is even, because $AAB$ or $ABB$ will never be prime.

$A=?$	$B=?$	$AAB=?$	$ABB=?$	$AABB=?$
$A=9$	$B=9$	$AAB=999$ ; not prime	N/A	N/A
$A=9$	$B=7$	$AAB=997$ ; prime	$ABB=997$ ; prime	$AABB=9977=11 \times 907$ ; not product of $3$ primes
$A=9$	$B=5$	$AAB=995$ ; not prime	N/A	N/A
$A=9$	$B=3$	$AAB=993$ ; not prime	N/A	N/A
$A=9$	$B=1$	$AAB=991$ ; prime	$ABB=991$ ; prime	$AABB=9911=11 \times 17 \times 53$ ; product of $3$ primes

Therefore, the largest number that fits all three requirements is $9911$ .

I messed up the values of the $ABB$ 's. It should be $977$ and $911$ respectively. Both are still prime, so the answer is still the same. I would edit the solution directly, but whenever I do, all my Latex goes away...

Joshua Lowrance - 2 years, 1 month ago

Find this number

The answer is 9911.

1 solution

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