## Task 3 - 230513

For the letters $a, b, c, d, e, f$, natural numbers are to be inserted in the following tasks (1) to (6) so that correct equations result. The same letters should represent the same numbers, and different letters should represent different numbers.

(1) $a+b+c=21$,

(2) $b \cdot c=42$,

(3) $c+d=70: b$,

(4) $e: a=d$,

(5) $c=54: 9$,

(6) $a+b+c+d+e+f=60$

Find such an assignment and verify that it meets all the requirements!
From (5) it follows that $\mathrm{c}=6$.

Substituting this into (2), we get $\mathrm{b} \cdot 6=42$, so $\mathrm{b}=42: 6$, i.e., $\mathrm{b}=7$.

From (1), it then follows that $\mathrm{a}+7+6=21$, so $\mathrm{a}=21-13$, i.e., $\mathrm{a}=8$.

From (3), we further obtain $\mathrm{d}+6=70: 7$, so $\mathrm{d}=10-6$, i.e., $\mathrm{d}=4$.

From (4), we then get $\mathrm{e} : 8=4$, so $\mathrm{e}=4 \cdot 8$, i.e., $\mathrm{e}=32$.

From (6), it finally follows that $8+7+6+4+32+\mathrm{f}=60$, so $\mathrm{f}=60-57$, i.e., $\mathrm{f}=3$.

The numbers $a, b, c, d, e, f$ found in this way are pairwise distinct and satisfy (1) to (6).