## Task 22/69

If the first digit of the birth and death years of a famous German scholar is omitted, two numbers $a$ and $b$ are obtained, for which the following applies:

1. Both numbers can be factored into a product of three different factors (all greater than 1), and among the 6 factors, 5 are prime numbers.
2. The two smaller factors of $a$ are each 2 less than the corresponding factors of $b$.
3. The largest factor of $a$ is equal to the sum of ten times the smallest factor and the middle factor of $a$.
4. The largest factor of $b$ is equal to the sum of twice the smallest factor and the middle factor of $b$, and is 1 greater than twice the middle factor.

Determine the two years and name the scholar!
The factors of $a$, in increasing order, are $x, y, z$, and the factors of $b$ are $u, v, w$. The following equations hold: $z=10 x+y, w=2 u+v=2 v+l$ or $z=10 x+y, v=2 u-1$. According to the last equation, the only natural numbers from 2 to 5 are possible for $u$, since 12 must be greater than 1, and for $u>5$, values for $v$ and $w$ would result in the product $uvw$ being greater than 1000.

The values 2 and 3 are excluded for $u$, as otherwise $x<2$ would be the case. For $u=4, v=7$, $w=15$ would result. However, then two factors would not be prime numbers. Therefore, only the solution $u=5, v=9, w=19$ is possible, from which $x=3, y=7$, and $z=37$ are derived.

The two years are thus 1777 and 1855. They are the birth year and the death year of Carl Friedrich Gauss.