Multiplying and Dividing by 1- and 2-Digit Numbers

1. Multiplying by a 1-digit number: partitioning

To multiply a big number by a single digit, partition the big number into hundreds, tens and ones, multiply each part, and add. This works because of the distributive property: 584 × 8 = (500 + 80 + 4) × 8 = 500 × 8 + 80 × 8 + 4 × 8.

Use your multiplication facts and powers of 10: 80 × 8 is 8 × 8 = 64 made ten times bigger, so 640. The short column method does the same thing from the right: 4 × 8 = 32 (write 2, carry 3), 8 × 8 = 64 plus 3 = 67 (write 7, carry 6), 5 × 8 = 40 plus 6 = 46. Answer: 4,672.

2. Multiplying by a 2-digit number

The place value (column) method. To find 345 × 27, think of 27 as 20 + 7. Multiply 345 by 7 to get 2,415. Then multiply 345 by 20: because the 2 is in the tens place, the row starts with a placeholder zero in the ones column, giving 6,900. Add the two partial products: 2,415 + 6,900 = 9,315.

▶ Notice the second partial product ends in a zero because the 2 in 27 is worth 20.

The lattice method. Draw a grid with one column for each digit of 345 and one row for each digit of 27, and draw a diagonal in each box. Write the product of each column digit and row digit in its box, tens above the diagonal and ones below. Then add along each diagonal strip, starting at the bottom right, carrying when needed.

▶ The lattice method for 345 × 27: multiply digit by digit, then add along the diagonals from the bottom right.

The distributive property. Sometimes the cleanest way is to split the 2-digit number into friendly parts. For 325 × 15, write 15 as 10 + 5: 325 × 15 = (325 × 10) + (325 × 5) = 3,250 + 1,625 = 4,875. Notice that 325 × 5 is half of 325 × 10, which makes it even quicker.

3. Dividing by 1- and 2-digit numbers

Division asks "how many groups of the divisor fit into the dividend?" We work from the left, the biggest place first. For short division by a single digit, such as 846 ÷ 6: 6 goes into 8 once, remainder 2; carry the 2 to make 24; 6 goes into 24 four times; 6 goes into 6 once. Answer: 141.

With a 2-digit divisor the same idea is written out in full as long division, so that you can see each subtraction. For 7,425 ÷ 15: 15 does not go into 7, so look at 74. 15 × 4 = 60, remainder 14. Bring down the 2 to make 142. 15 × 9 = 135, remainder 7. Bring down the 5 to make 75. 15 × 5 = 75, remainder 0. Answer: 495.

▶ Each step is: divide, multiply, subtract, bring down. Watch the remainder and the next digit join together.

1. 1Divide: how many times does the divisor go into the current number?
2. 2Multiply: write the quotient digit on top and multiply it by the divisor.
3. 3Subtract: find the remainder. It must be smaller than the divisor.
4. 4Bring down the next digit and repeat until no digits are left.
5. 5Check: quotient × divisor + remainder = dividend.

4. Divisibility tests

A divisibility test tells you whether a division will come out exactly, without doing it. These tests help you simplify fractions, find factors and check answers.

The test for 7 can be repeated on big numbers. Is 2,814 divisible by 7? Double the last digit: 4 × 2 = 8; subtract from the rest: 281 − 8 = 273. Still big, so repeat: 3 × 2 = 6; 27 − 6 = 21. 21 = 3 × 7, so 2,814 is divisible by 7 (in fact 2,814 ÷ 7 = 402).

Worked examples

Try each one yourself first, then open the solution and compare every step.