Multiply by 11 and 101

The idea

Since 11 = 10 + 1, multiplying a number by 11 adds it to itself shifted one place, so dcba × 11 comes from the column sums d, c+d, b+c, a+b, a; for 101 = 100 + 1 the shift is two places.

Break 11 into 10 + 1

Use 11 = 10 + 1: 3874 × 11 = 3874 × (10 + 1) = 38740 + 3874

Add them. What is 3874 × 11?

• 41614
• 42514
• 42614

Add the columns

The number is written twice. The top copy is shifted one place left (× 10). Tap each column to add its two digits.

Shift and add

Since 11 = 10 + 1, multiplying by 11 adds the number to itself shifted one place. For dcba × 11 the column sums are d, c + d, b + c, a + b, a.

For 101 = 100 + 1 the shift is two places. The column sums of dcba × 101 are d, c, b + d, a + c, b, a.

3874 × 11 has column sums 3, 11, 15, 11, 4. Carry, and read 42614.

Notes

Multiplying by 11 = 10 + 1 adds the number to itself shifted one place; by 101 = 100 + 1, shifted two places. Add column by column and carry.

Check yourself

Use column sums to find 2354 × 11.

Answer: 25894

Sums: 2, 3 + 2 = 5, 5 + 3 = 8, 4 + 5 = 9, 4. Read 25894. Check: 2354 × 11 = 25894.

Find 6275 × 11. One column sum is 10 or more, so carry.

Answer: 69025

From the right: 5; 12 → write 2, carry 1; 9 + 1 = 10 → write 0, carry 1; 8 + 1 = 9; 6. Read 69025.

Now multiply by 101. Use the column sums to find 2345 × 101.

Answer: 236845

Sums: 2, 3, 4 + 2 = 6, 5 + 3 = 8, 4, 5. Read 236845. Check: 2345 × 101 = 236845.

Why is the number shifted two places when we multiply by 101?

• 101 = 10 + 1, so the shift is two places. 101 is not 10 + 1. That would be 11.
• 101 = 100 + 1, and × 100 shifts the number two places — correct. Yes! dcba × 101 = dcba × 100 + dcba. Two zeros, so the shift is two places.
• Because 101 has three digits. The reason is 101 = 100 + 1, not the number of digits.