Largest square factor

Number Theory Level 3

What is the maximum value of $a$ if ${ a }^{ 2 }$ is a factor of $(10\times 11\times 12\times ...\times 19)$ ?

1 solution

Arulx Z
Dec 9, 2015

I did the calculations by manually listing out the factors (I ignored primes larger than 11 because $11 \times 2 > 19$ ).

Prime factorization -

$2^9 \cdot 3^4 \cdot 5^2 \cdot 7$

We can subtract 1 from the odd powers to get -

$a^2=2^8\cdot 3^4 \cdot 5^2 \\ a = 2^4 \cdot 3^2 \cdot 5 = 720$

Good approach working with the prime factorization to find the "square factor".

why subtract 1 from the odd power?

Vishal Arya - 4 years, 2 months ago