2. A natural number is called " $p$ lucky" if it has $p$ digits and is divisible by $p$.

a) What is the smallest "13 lucky" number? (Justify your answer)

b) What is the largest "13 lucky" number? (Justify your answer)
Subject 2, author Prof. Dana Radu

A natural number is called " $p$ lucky " if it has $p$ digits and is divisible by $p$.

a) What is the smallest "13 lucky" number?

b) What is the largest "13 lucky" number?

| Solution Details | Associated Scoring |
| :--- | :---: |
| a) A= $\overline{a b c a b c}=\overline{a b c} \cdot 1001=\overline{a b c} \cdot 7 \cdot 11 \cdot 13: 13$ | 1 |
| $\mathrm{B}=999999999999=\left(999999 \cdot 10^{6}+999999\right): 13(\mathrm{~B}$ has 12 digits $)$ | 1 |
| The smallest 13 lucky number is $\mathrm{B}+13=100000000012$ | 2 |
| b) $10 \cdot \mathrm{B}+13$ has 14 digits | 1 |
| The largest lucky number is thus $10 \mathrm{~B}=9999999999990$ | 2 |