# Biggest Number

**Number Theory**Level 3

We have a 10-digit number (the first digit is not zero) represented by ABCDEFGHIJ wherein each digit \( \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \} \) is used exactly once.

Given the following conditions, what is the number?

1) A + B + C + D + E is a multiple by 6.

2) F + G + H + I + J is a multiple by 5.

3) A + C + E + G + I is a multiple by 9.

4) B + D + F + H + J is a multiple by 2.

5) AB is a multiple by 3.*

6) CD is a multiple by4.*

7) EF is a multiple by 7.*

8) GH is a multiple by 8.*

9) IJ is a multiple by 10.*

10) FE, HC and JA are all prime numers.*

*This is not multiplication, but refers to concatenation. For example, if \(A=1, B=2 \), then \(AB = 12 \).