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min integer, related to sum of digits

Source: 2006 VMEO III Shortlist SL N14 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2461015_vmeo__viet

October 28, 2021
Digitssum of digitsnumber theory

Problem Statement

For any natural number n=ai...a2a1‾n = \overline{a_i...a_2a_1}, consider the number T(n)=10∑i  evenai+∑i  oddai.T(n) =10 \sum_{i \,\, even} a_i+\sum_{i \,\, odd} a_i. Let's find the smallest positive integer AA such that is sum of the natural numbers n1,n2,...,n148n_1,n_2,...,n_{148} as well as of m1,m2,...,m149m_1,m_2,...,m_{149} and matches the pattern:
A=n1+n2+...+n148=m1+m2+...+m149A=n_1+n_2+...+n_{148}=m_1+m_2+...+m_{149} T(n1)=T(n2)=...=T(n148)T(n_1)=T(n_2)=...=T(n_{148}) T(m1)=T(m2)=...=T(m148)T(m_1)=T(m_2)=...=T(m_{148})
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