## Task 2 - 200822

Determine all pairs $(a ; b)$ of natural numbers with $a<b$ that have the following properties:

The sum of the numbers $a$ and $b$ is 192.

The greatest common divisor of the numbers $a$ and $b$ is 24.
If $(a ; b)$ is a pair of natural numbers with the required properties, it follows:

24 is a divisor of both $a$ and $b$, so there exist natural numbers $p, q$ such that $a=24 p, b=24 q$. Since 24 is even the greatest common divisor of $a$ and $b$, it further follows that $p$ and $q$ are coprime. (1)

From $a<b$ it follows that $24 p<24 q$, thus $p<q ; \quad$ (2)

from $a+b=192$ it follows that $24 p+24 q=192$, thus $p+q=8$.

The requirements (2) and (3) are only met by the following natural numbers $p, q$:

| $p$ | $q$ |
| :---: | :---: |
| 0 | 8 |
| 1 | 7 |
| 2 | 6 |
| 3 | 5 |

Requirement (1) is only satisfied for $p=1, q=7$ and for $p=3, q=5$. Therefore, only the pairs $(24 ; 168),(72 ; 120)$ can have the required properties.

These pairs do indeed have these properties; because:

$24<168,72<120 ; \quad 24+168=192,72+120=192$.

Since $168=7 \cdot 24$, 24 is the gcd of 24 and 168, and since $72=3 \cdot 24$ and $120=5 \cdot 24$, 24 is the gcd of 72 and 120, as 3 and 5 are coprime.