## Task 3 - 200723

Jens says: "I am thinking of two natural numbers. Their least common multiple (LCM) is 51975, their greatest common divisor (GCD) is 45. One of the numbers is 4725."

Determine whether there is exactly one natural number that can be the second number Jens is thinking of! If this is the case, find this second number!
The prime factorizations of the mentioned numbers are:

$\mathrm{kgV}: 51975=33 \cdot 52 \cdot 7 \cdot 11$,

$\operatorname{ggT}: 45=32 \cdot 5$,

first number: $4725=33 \cdot 52 \cdot 7$.

If a natural number $z$ can be the second thought number, then for it:

In its prime factorization, it can contain at most those prime numbers that appear in the $\mathrm{kgV}$, i.e., at most the prime numbers 3, 5, 7, 11. The prime number 3 appears in 4725 in a greater quantity than in the $\operatorname{ggT}$, where it appears in the quantity 2. Therefore, it must appear in $z$ in this quantity 2 as a factor.

The prime number 5 appears in 4725 in a greater quantity than in the $\operatorname{ggT}$, where it appears in the quantity 1. Therefore, it must appear in $z$ in this quantity 1 as a factor.

The prime number 7 appears in 4725 but not in the $\operatorname{ggT}$. Therefore, it cannot appear in $z$. The prime number 11 does not appear in 4725 but does appear in the $\mathrm{kgV}$, where it appears in the quantity 1. Therefore, it must appear in $z$ in this quantity 1 as a factor.

Thus, the number $z=32 \cdot 5 \cdot 11=495$ can be the second thought number at most. It can actually be this; because $4725=33 \cdot 52 \cdot 7$ and $z=32 \cdot 5 \cdot 11$ have the $\mathrm{kgV} 33 \cdot 52 \cdot 7 \cdot 11=51975$ and the $\operatorname{ggT~} 32 \cdot 5=45$. Therefore, there is exactly one natural number that can be the second thought number: it is 495.