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\begin{array}{c}\phantom{\times}1512258\\\underline{\times\phantom{}2563}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times}1512258\\\underline{\times\phantom{}2563}\\\phantom{\times}4536774\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 1512258 with 3. Write the result 4536774 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times}1512258\\\underline{\times\phantom{}2563}\\\phantom{\times}4536774\\\phantom{\times}9073548\phantom{9}\\\end{array}
Now multiply the first number with the 2^{nd} digit in 2^{nd} number to get intermediate results. That is Multiply 1512258 with 6. Write the result 9073548 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times}1512258\\\underline{\times\phantom{}2563}\\\phantom{\times}4536774\\\phantom{\times}9073548\phantom{9}\\\phantom{\times}7561290\phantom{99}\\\end{array}
Now multiply the first number with the 3^{rd} digit in 2^{nd} number to get intermediate results. That is Multiply 1512258 with 5. Write the result 7561290 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times}1512258\\\underline{\times\phantom{}2563}\\\phantom{\times}4536774\\\phantom{\times}9073548\phantom{9}\\\phantom{\times}7561290\phantom{99}\\\underline{\phantom{\times}3024516\phantom{999}}\\\end{array}
Now multiply the first number with the 4^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 1512258 with 2. Write the result 3024516 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times}1512258\\\underline{\times\phantom{}2563}\\\phantom{\times}4536774\\\phantom{\times}9073548\phantom{9}\\\phantom{\times}7561290\phantom{99}\\\underline{\phantom{\times}3024516\phantom{999}}\\\phantom{\times}-419050042\end{array}
Now add the intermediate results to get final answer.