Long Multiplication

Multiply When the Multiplier Has a Zero Digit

Multiply 143 by 30. The ones digit is 0, so the entire ones-row partial product is 0. Only the tens-digit partial product matters; it shifts one column left. This lesson shows why multiplying by a multiple of 10 just appends a zero and then multiplies the rest.

zero-partial-product When a multiplier digit is 0, its partial product row is all zeros. You can skip the row and just shift the next partial left.
multiply-by-10 Shifting one column left is the same as multiplying by 10.

Example

Multiply 143 by 30 and keep the zero partial clear.

highlighted = computed this step

Step 1 — Set up

Set up the multiplication board.

xx143×xx30xxxx□x□□□□x□□□□\begin{array}{rrrrr}\phantom{x}&\phantom{x}&1&4&3\\\times&\phantom{x}&\phantom{x}&3&0\\\hline\\\phantom{x}&\phantom{x}&\phantom{x}&\phantom{x}&\square\\\phantom{x}&\square&\square&\square&\square\\\hline\\\phantom{x}&\square&\square&\square&\square\end{array}
Set up 143 x 30aligned partials--143x----30------------------------zero ones row, shifted tens row, then sumones row: x 0143 x 0 = 0The ones digit of 30 is 0, so this row contributes 0.tens row: x 3 tens3 x 3 = 9, write 9 carry 03 x 4 tens + carry 0 = 12 tens, write 2 carry 13 x 1 hundreds + carry 1 = 4 hundredsraw 429, shift one place to 4290add aligned partials-- + -- = --

Step 2 — Ones partial

Ones digit 0: 143 x 0 = 0, write the partial.

xx143×xx30xxxx0x□□□□x□□□□\begin{array}{rrrrr}\phantom{x}&\phantom{x}&1&4&3\\\times&\phantom{x}&\phantom{x}&3&0\\\hline\\\phantom{x}&\phantom{x}&\phantom{x}&\phantom{x}&\hl{0}\\\phantom{x}&\square&\square&\square&\square\\\hline\\\phantom{x}&\square&\square&\square&\square\end{array}
ones digit is 0: 143 x 0 = 0aligned partials--143x----30------0----------------zero ones row, shifted tens row, then sumones row: x 0143 x 0 = 0The ones digit of 30 is 0, so this row contributes 0.tens row: x 3 tens3 x 3 = 9, write 9 carry 03 x 4 tens + carry 0 = 12 tens, write 2 carry 13 x 1 hundreds + carry 1 = 4 hundredsraw 429, shift one place to 4290add aligned partials-- + -- = --

Step 3 — Tens partial

Tens digit 3: 4 x 3 = 12, write 2, carry 1; shift partial to 4290.

xx1xxxx143×xx30xxxx0x4290x□□□□\begin{array}{rrrrr}\phantom{x}&\phantom{x}&\hl{1}&\phantom{x}&\phantom{x}\\\phantom{x}&\phantom{x}&1&4&3\\\times&\phantom{x}&\phantom{x}&3&0\\\hline\\\phantom{x}&\phantom{x}&\phantom{x}&\phantom{x}&0\\\phantom{x}&\hl{4}&\hl{2}&\hl{9}&\hl{0}\\\hline\\\phantom{x}&\square&\square&\square&\square\end{array}
3 tens x 143 gives 4290aligned partials01carries--143x----30------04290--------zero ones row, shifted tens row, then sumones row: x 0143 x 0 = 0The ones digit of 30 is 0, so this row contributes 0.tens row: x 3 tens3 x 3 = 9, write 9 carry 03 x 4 tens + carry 0 = 12 tens, write 2 carry 13 x 1 hundreds + carry 1 = 4 hundredsraw 429, shift one place to 4290add aligned partials-- + -- = --

Step 4 — Add partials

Add partials: 0 + 4290 = 4290.

xx143×xx30xxxx0x4290x4290\begin{array}{rrrrr}\phantom{x}&\phantom{x}&1&4&3\\\times&\phantom{x}&\phantom{x}&3&0\\\hline\\\phantom{x}&\phantom{x}&\phantom{x}&\phantom{x}&0\\\phantom{x}&4&2&9&0\\\hline\\\phantom{x}&\hl{4}&\hl{2}&\hl{9}&\hl{0}\end{array}
0 + 4290 = 4290aligned partials--143x----30------042904290zero ones row, shifted tens row, then sumones row: x 0143 x 0 = 0The ones digit of 30 is 0, so this row contributes 0.tens row: x 3 tens3 x 3 = 9, write 9 carry 03 x 4 tens + carry 0 = 12 tens, write 2 carry 13 x 1 hundreds + carry 1 = 4 hundredsraw 429, shift one place to 4290add aligned partials0 + 4290 = 4290

Step 5 — Result

Read the complete product: 4290.

xx143×xx30xxxx0x4290x4290\begin{array}{rrrrr}\phantom{x}&\phantom{x}&1&4&3\\\times&\phantom{x}&\phantom{x}&3&0\\\hline\\\phantom{x}&\phantom{x}&\phantom{x}&\phantom{x}&0\\\phantom{x}&4&2&9&0\\\hline\\\phantom{x}&4&2&9&0\end{array}
143 x 30 = 4290aligned partials--143x----30------042904290zero ones row, shifted tens row, then sumones row: x 0143 x 0 = 0The ones digit of 30 is 0, so this row contributes 0.tens row: x 3 tens3 x 3 = 9, write 9 carry 03 x 4 tens + carry 0 = 12 tens, write 2 carry 13 x 1 hundreds + carry 1 = 4 hundredsraw 429, shift one place to 4290add aligned partials0 + 4290 = 4290