Multiply 234 by 56 with shifted partial products.
Step 1 — Set up
Set up the multiplication board.
x x x 2 3 4 × x x x 5 6 x x □ □ □ □ x □ □ □ □ □ x □ □ □ □ □ \begin{array}{rrrrrr}\phantom{x}&\phantom{x}&\phantom{x}&2&3&4\\\times&\phantom{x}&\phantom{x}&\phantom{x}&5&6\\\hline\\\phantom{x}&\phantom{x}&\square&\square&\square&\square\\\phantom{x}&\square&\square&\square&\square&\square\\\hline\\\phantom{x}&\square&\square&\square&\square&\square\end{array} x × x x x x x x □ □ x x □ □ □ 2 x □ □ □ 3 5 □ □ □ 4 6 □ □ □ Set up 234 x 56 aligned partials -- -- 2 3 4 x -- -- -- 5 6 -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- ones row, shifted tens row, then sum ones row: x 6 6 x 4 = 24, write 4 carry 2 6 x 3 tens + carry 2 = 20 tens, write 0 carry 2 6 x 2 hundreds + carry 2 = 14 hundreds; row 1404 tens row: x 5 tens 5 x 4 = 20, write 0 carry 2 5 x 3 tens + carry 2 = 17 tens, write 7 carry 1 5 x 2 hundreds + carry 1 = 11 hundreds raw 1170, shift one place to 11700 add aligned partials -- + -- = --
Step 2 — Ones partial
Ones digit 6: 4 x 6 = 24, write 4, carry 2; partial 1404.
x x x 2 2 x x x x 2 3 4 × x x x 5 6 x x 1 4 0 4 x □ □ □ □ □ x □ □ □ □ □ \begin{array}{rrrrrr}\phantom{x}&\phantom{x}&\phantom{x}&\hl{2}&\hl{2}&\phantom{x}\\\phantom{x}&\phantom{x}&\phantom{x}&2&3&4\\\times&\phantom{x}&\phantom{x}&\phantom{x}&5&6\\\hline\\\phantom{x}&\phantom{x}&\hl{1}&\hl{4}&\hl{0}&\hl{4}\\\phantom{x}&\square&\square&\square&\square&\square\\\hline\\\phantom{x}&\square&\square&\square&\square&\square\end{array} x x × x x x x x x x □ □ x x x 1 □ □ 2 2 x 4 □ □ 2 3 5 0 □ □ x 4 6 4 □ □ 6 x 4 = 24: write 4, carry 2 aligned partials 2 2 carries -- -- 2 3 4 x -- -- -- 5 6 -- 1 4 0 4 -- -- -- -- -- -- -- -- -- -- ones row, shifted tens row, then sum ones row: x 6 6 x 4 = 24, write 4 carry 2 6 x 3 tens + carry 2 = 20 tens, write 0 carry 2 6 x 2 hundreds + carry 2 = 14 hundreds; row 1404 tens row: x 5 tens 5 x 4 = 20, write 0 carry 2 5 x 3 tens + carry 2 = 17 tens, write 7 carry 1 5 x 2 hundreds + carry 1 = 11 hundreds raw 1170, shift one place to 11700 add aligned partials -- + -- = --
Step 3 — Tens partial
Tens digit 5: 4 x 5 = 20, write 0, carry 2; shift partial to 11700.
x x x 1 2 x x x x 2 3 4 × x x x 5 6 x x 1 4 0 4 x 1 1 7 0 0 x □ □ □ □ □ \begin{array}{rrrrrr}\phantom{x}&\phantom{x}&\phantom{x}&\hl{1}&\hl{2}&\phantom{x}\\\phantom{x}&\phantom{x}&\phantom{x}&2&3&4\\\times&\phantom{x}&\phantom{x}&\phantom{x}&5&6\\\hline\\\phantom{x}&\phantom{x}&1&4&0&4\\\phantom{x}&\hl{1}&\hl{1}&\hl{7}&\hl{0}&\hl{0}\\\hline\\\phantom{x}&\square&\square&\square&\square&\square\end{array} x x × x x x x x x x 1 □ x x x 1 1 □ 1 2 x 4 7 □ 2 3 5 0 0 □ x 4 6 4 0 □ 5 tens x 234 gives 11700 aligned partials 2 1 carries -- -- 2 3 4 x -- -- -- 5 6 -- 1 4 0 4 1 1 7 0 0 -- -- -- -- -- ones row, shifted tens row, then sum ones row: x 6 6 x 4 = 24, write 4 carry 2 6 x 3 tens + carry 2 = 20 tens, write 0 carry 2 6 x 2 hundreds + carry 2 = 14 hundreds; row 1404 tens row: x 5 tens 5 x 4 = 20, write 0 carry 2 5 x 3 tens + carry 2 = 17 tens, write 7 carry 1 5 x 2 hundreds + carry 1 = 11 hundreds raw 1170, shift one place to 11700 add aligned partials -- + -- = --
Step 4 — Add partials
Add partials: 1404 + 11700 = 13104.
x x x 2 3 4 × x x x 5 6 x x 1 4 0 4 x 1 1 7 0 0 x 1 3 1 0 4 \begin{array}{rrrrrr}\phantom{x}&\phantom{x}&\phantom{x}&2&3&4\\\times&\phantom{x}&\phantom{x}&\phantom{x}&5&6\\\hline\\\phantom{x}&\phantom{x}&1&4&0&4\\\phantom{x}&1&1&7&0&0\\\hline\\\phantom{x}&\hl{1}&\hl{3}&\hl{1}&\hl{0}&\hl{4}\end{array} x × x x x x x x 1 1 x x 1 1 3 2 x 4 7 1 3 5 0 0 0 4 6 4 0 4 1404 + 11700 = 13104 aligned partials -- -- 2 3 4 x -- -- -- 5 6 -- 1 4 0 4 1 1 7 0 0 1 3 1 0 4 ones row, shifted tens row, then sum ones row: x 6 6 x 4 = 24, write 4 carry 2 6 x 3 tens + carry 2 = 20 tens, write 0 carry 2 6 x 2 hundreds + carry 2 = 14 hundreds; row 1404 tens row: x 5 tens 5 x 4 = 20, write 0 carry 2 5 x 3 tens + carry 2 = 17 tens, write 7 carry 1 5 x 2 hundreds + carry 1 = 11 hundreds raw 1170, shift one place to 11700 add aligned partials 1404 + 11700 = 13104
Step 5 — Result
Read the complete product: 13104.
x x x 2 3 4 × x x x 5 6 x x 1 4 0 4 x 1 1 7 0 0 x 1 3 1 0 4 \begin{array}{rrrrrr}\phantom{x}&\phantom{x}&\phantom{x}&2&3&4\\\times&\phantom{x}&\phantom{x}&\phantom{x}&5&6\\\hline\\\phantom{x}&\phantom{x}&1&4&0&4\\\phantom{x}&1&1&7&0&0\\\hline\\\phantom{x}&1&3&1&0&4\end{array} x × x x x x x x 1 1 x x 1 1 3 2 x 4 7 1 3 5 0 0 0 4 6 4 0 4 234 x 56 = 13104 aligned partials -- -- 2 3 4 x -- -- -- 5 6 -- 1 4 0 4 1 1 7 0 0 1 3 1 0 4 ones row, shifted tens row, then sum ones row: x 6 6 x 4 = 24, write 4 carry 2 6 x 3 tens + carry 2 = 20 tens, write 0 carry 2 6 x 2 hundreds + carry 2 = 14 hundreds; row 1404 tens row: x 5 tens 5 x 4 = 20, write 0 carry 2 5 x 3 tens + carry 2 = 17 tens, write 7 carry 1 5 x 2 hundreds + carry 1 = 11 hundreds raw 1170, shift one place to 11700 add aligned partials 1404 + 11700 = 13104