Multiplying two 2-digit numbers
(same 1st digit)
1. Select two 2-digit numbers with the same first digit.
2. Multiply their second digits (keep the carry). _ _ _ X
3. Multiply the sum of the second digits by the first digit,
add the carry (keep the carry). _ _ X _
4. Multiply the first digits (add the carry). X X _ _
Example:
1. If the first number is 42, choose 45 as the second number (any 2-digit number with first digit 4).
2. Multiply the last digits: 2 × 5 = 10 (keep carry)
_ _ _ 0
3. Multiply the sum of the 2nd digits by the first:
5 + 2 = 7; 7 × 4 = 28; 28 + 1 = 29 (keep carry)
_ _ 9 _
4. Multiply the first digits (add the carry)
4 × 4 = 16; 16 + 2 = 18
1 8 _ _
5. So 42 × 45 = 1890.
See the pattern?
1. If the first number is 62, choose 67 as the second number
(any 2-digit number with first digit 6).
2. Multiply the last digits: 2 × 7 = 14 (keep carry)
_ _ _ 4
3. Multiply the sum of the 2nd digits by the first (add carry):
2 + 7 = 9; 6 × 9 = 54; 54 + 1 = 55 (keep carry)
_ _ 5 _
4. Multiply the first digits (add the carry)
6 × 6 = 36; 36 + 5 = 41
4 1 _ _
5. So 62 × 67 = 4154.
Multiplying two 2-digit numbers
(same 1st digit, 2nd digits sum to 10)
1. Both numbers should have the same first digit.
2. Choose second digits whose sum is 10.
3. Multiply the first digit by one number greater than itself; this number will be the first part of the answer:
X X _ _.
4. Multiply the two second digits together; the product
will be the last part of the answer: _ _ X X.
Note: If the two second digits are 1 and 9 (or, more generally, have a product that is less than ten), insert a 0 (zero) for the first X in step 4.
(Thanks to Michael Richardson, age 10, for this note.)
Example:
1. If the first number is 47, choose 43 as the second number (same first digit, second digits add to 10).
2. 4 × 5 = 20 (multiply the first digit by one number greater than itself): the first part of the answer is
2 0 _ _.
3. 7 × 3 = 21 (multiply the two second digits together); the last part of the answer is _ _ 2 1.
4. So 47 × 43 = 2021.
See the pattern?
1. If the first number is 62, choose 68 as the second number (same first digit, second digits add to 10).
2. 6 × 7 = 42 (multiply the first digit by one greater), the first part of the answer is 4 2 _ _.
3. 2 × 8 = 16 (multiply the two second digits together); the last part of the answer is _ _ 1 6.
4. So 62 × 68 = 4216.
Multiplying two 2-digit numbers
(same 2nd digit)
1. Both numbers should have the same second digit.
2. Choose first digits whose sum is 10.
3. Multiply the first digits and add one second: X X _ _.
4. Multiply the second digits together: _ _ X X.
Example:
1. If the first number is 67, choose 47 as the second number (same second digit, first digits add to 10).
2. Multiply the 1st digits, add one 2nd.
6x4 = 24, 24+7 = 31. 3 1 _ _
3. Multiply the 2nd digits. 7x7 = 49 _ _ 4 9
4. So 67 × 47 = 3149.
See the pattern?
1. If the first number is 93, choose 13 as the second number (same second digit, first digits add to 10).
2. Multiply the 1st digits, add one 2nd. 9x1 = 9, 9+3 = 12.
1 2 _ _
3. Multiply the 2nd digits. 3x3 = 9 _ _ 0 9
4. So 93 × 13 = 1209.
Multiplying two selected 3-digit numbers
(middle digit 0)
1. Select a 3-digit number with a middle digit of 0.
2. Choose a multiplier with the same first two digits, whose third digit sums to 10 with the third digit of the first 3-digit number.
3. The first digit(s) will be the square of the first digit:
X _ _ _ _ or X X _ _ _ _.
4. The next digit will be the first digit of the numbers:
_ X _ _ _ or _ _ X _ _ _.
5. The next digit is zero: _ _ 0 _ _ or _ _ _ 0 _ _.
6. The last two digits will be the product of the third digits:
_ _ _ X X or _ _ _ _ X X.
Example:
1. If the first number is 407, choose 403 as the second number (same first digits, second digits add to 10).
2. 4 × 4 = 16 (square the first digit): 1 6 _ _ _ _.
3. The next digit will be the first digit of the numbers:
_ _ 4 _ _ .
4. The next digit is zero: _ _ 0 _ _ .
5. 7 × 3 = 21 (the last two digits will be the product of the third digits: _ _ _ 2 1.
6. So 407 × 403 = 164021.
See the pattern?
1. If the first number is 201, choose 209 as the second number (same first digits, second digits add to 10).
2. 2 × 2 = 4 (square the first digit): 4 _ _ _ _.
3. The next digit will be the first digit of the numbers:
_ 2 _ _ _ .
4. The next digit is zero: _ _ 0 _ _ .
5. 1 × 9 = 09 (the last two digits will be the product of the third digits: _ _ _ 0 9.
6. So 201 × 209 = 42009.
Multiplying two selected 3-digit numbers
(middle digit 1)
1. Select a 3-digit number with a middle digit of 1.
2. Choose a multiplier with the same first two digits, whose third digit sums to 10 with the third digit of the first 3-digit number.
3. The last two digits will be the product of the first digits:
_ _ _ 0 X or _ _ _ _ X X.
4. The third digit from the right will be 2:
_ _ 2 _ _ or _ _ _ 2 _ _ .
5. The next digit to the left will be 3 times the first digit of the number (keep carry):
_ X _ _ _ or _ _ X _ _ _.
6. The first digits will be the square of the first digit plus the carry:
X _ _ _ _ or X X _ _ _ _.
As you determine the digits in the answer from right to left, repeat them to yourself at each step until you have the whole answer.
Example:
1. If the first number is 814, choose 816 as the second number (same first digits, second digits add to 10).
2. 4 × 6 = 24 (multiply the first digits) - last two digits:
_ _ _ _ 2 4.
3. The third digit from the right is 2: _ _ 2 _ _ .
4. 8 × 3 = 24 (the next digit to the left is 3 times the first digit (keep carry 2): _ _ 4 _ _ _ .
5. 8 × 8 = 64; 64 + 2 = 66 ( the first digits are the square of the first digit plus the carry: 6 6 _ _ _ _.
6. So 814 × 816 = 664224.
See the pattern?
1. If the first number is 317, choose 313 as the multiplier (same first digits, second digits add to 10).
2. 7 × 3 = 21 (multiply the first digits) - last two digits:
_ _ _ _ 2 1.
3. The third digit from the right is 2: _ _ 2 _ _ .
4. 3 × 3 = 9 (the next digit to the left is 3 times the first digit (no carry): _ 9 _ _ _ .
5. 3 × 3 = 9 ( the first digits are the square of the first digit: 9 _ _ _ _.
6. So 317 × 313 = 99221.
Multiplying two selected 3-digit numbers
(middle digit 2)
1. Select a 3-digit number with a middle digit of 2.
2. Choose a multiplier with the same first two digits, whose third digit sums to 10 with the third digit of the first 3-digit number.
3. The last two digits will be the product of the third digits: _ _ _ _X X.
4. The third digit from the right will be 6: _ _ 6 _ _.
5. The next digit to the left will be 5 times the first digit of the number (keep carry): _ _ X _ _ _.
6. The first digits will be the square of the first digit plus the carry: X X _ _ _ _.
As you determine the digits in the answer from right to left, repeat them to yourself at each step until you have the whole answer.
Example:
1. If the first number is 622, choose 628 as the second number (same first digits, third digits add to 10).
2. 2 × 8 = 16 (multiply the third digits) - last two digits:
_ _ _ _ 1 6.
3. The third digit from the right is 6: _ _ 6 _ _ _.
4. 5 × 6 = 30 (the next digit to the left is 5 times the first digit (keep carry 3): _ _ 0 _ _ _ .
5. 6 × 6 = 36; 64 + 3 = 39 ( the first digits are the square of the first digit plus the carry: 3 9 _ _ _ _.
6. So 622 × 628 = 390616.
See the pattern?
1. If the first number is 221, choose 229 as the second number (same first digits, third digits add to 10).
2. 1 × 9 = 9 (multiply the third digits) - last two digits:
_ _ _ 0 9.
3. The third digit from the right is 6: _ _ 6 _ _.
4. 5 × 2 = 10 (the next digit to the left is 5 times the first digit (keep carry 1): _ 0 _ _ _ .
5. 2 × 2 = 4; 4 + 1 = 5 ( the first digits are the square of the first digit plus the carry: 5 _ _ _ _.
6. So 221 × 229 = 50609.
Multiplying two selected 3-digit numbers
(middle digit 3)
1. Select a 3-digit number with a middle digit of 3 (last digit not zero).
2. Choose a multiplier with the same first two digits, whose third digit sums to 10 with the third digit of the first 3-digit number.
3. The last two digits will be the product of the third digits: _ _ _ _ X X.
4. The third digit from the right will be 2: _ _ 2 _ _.
5. The next digit to the left will be 7 times the first digit of the number plus 1 (keep carry): _ _ X _ _ _.
6. The first digits will be the square of the first digit plus the carry: X X _ _ _ _.
As you determine the digits in the answer from right to left, repeat them to yourself at each step until you have the whole answer.
Example:
1. If the first number is 631, choose 639 as the second number (same first digits, third digits add to 10).
2. 1 × 9 = 09 (multiply the third digits) - last two digits:
_ _ _ _ 0 9.
3. The third digit from the right is 2: _ _ 2 _ _ _.
4. 7 × 6 = 42, 42 + 1 = 43 (the next digit to the left is 7 times the first digit plus 1 (keep carry 4): _ _ 3 _ _ _ .
5. 6 × 6 = 36; 36 + 4 = 40 ( the first digits are the square of the first digit plus the carry: 4 0 _ _ _ _.
6. So 631 × 639 = 403209.
See the pattern?
1. If the first number is 236, choose 234 as the second number (same first digits, third digits add to 10).
2. 6 × 4 = 24 (multiply the third digits) - last two digits: _ _ _ _ 2 4.
3. The third digit from the right is 2: _ _ 2 _ _ _.
4. 7 × 2 = 14, 14 + 1 = 15 (the next digit to the left is 7 times the first digit plus 1 (keep carry 1): _ _ 5 _ _ _ .
5. 2 × 2 = 4; 4 + 1 = 5 ( the first digits are the square of the first digit plus the carry: 0 5 _ _ _ _.
6. So 236 × 234 = 55224.
Multiplying two selected 3-digit numbers
(middle digit 4)
1. Select a 3-digit number with a middle digit of 4 (last digit not zero).
2. Choose a multiplier with the same first two digits, whose third digit sums to 10 with the third digit of the first 3-digit number.
3. The last three digits will be 0 and the product of the third digits: _ _ _ 0 X X.
4. The third digit from the right will be 9 times the first digit + 2 (keep the carry): _ _ X _ _.
5. The first two digits will be the square of the first digit plus the carry: X X _ _ _ _.
As you determine the digits in the answer from right to left, repeat them to yourself at each step until you have the whole answer.
Example:
1. If the first number is 541, choose 549 as the second number (same first digits, third digits add to 10).
2. Last three digits: 0 and the product of the third digits: 1 × 9 = 9: _ _ 0 0 9
3. Next digit: 9 times the first digit + 2: 9 × 5 = 45, 45 + 2 = 47 (keep carry 4): _ _ 7 _ _ _
4. First two digits: square the first and carry: 5 × 5 = 25, 25 + 4 = 29: 2 9 _ _ _ _
5. So 541 × 549 = 297009.
See the pattern?
1. If the first number is 344, choose 346 as the second number (same first digits, third digits add to 10).
2. Last three digits: 0 and the product of the third digits: 4 × 6 = 24: _ _ 0 2 4
3. Next digit: 9 times the first digit + 2: 9 × 3 = 27, 27 + 2 = 29 (keep carry 2): _ _ 9 _ _ _
4. First two digits: square the first and carry: 3 × 3 = 9, 9 + 2 = 11: 1 1 _ _ _ _
5. So 344 × 346 = 119024.
Multiplying two selected 3-digit numbers
(middle digit 5)
1. Select a 3-digit number with a middle digit of 5 (last digit not zero).
2. Choose a multiplier with the same first two digits, whose third digit sums to 10 with the third digit of the first 3-digit number.
3. The last three digits will be 0 and the product of the third digits: _ _ _ 0 X X.
4. The third digit from the right will be the first digit + 3 (keep the carry): _ _ X _ _.
5. The first digit will be the first digit times the next number plus the carry: X X _ _ _ _.
Repeat those digits from left to right as you get them.
Example:
1. If the first number is 752, choose 758 as the second number (same first digits, third digits add to 10).
2. Last three digits: 0 and the product of the third digits:
2 × 8 = 16: _ _ 0 1 6
3. Next digit: first digit + 3: 7 + 3 = 10 (keep carry 1):
_ _ 0 _ _ _
4. First two digits: first digit times next number plus carry: 7 × 8 = 56, 56 + 1 = 57: 5 7 _ _ _ _
5. So 752 × 758 = 570016.
See the pattern?
1. If the first number is 654, choose 656 as the second number (same first digits, third digits add to 10).
2. Last three digits: 0 and the product of the third digits:
4 × 6 = 24: _ _ 0 2 4
3. Next digit: first digit + 3: 6 + 3 = 9: _ _ 9 _ _ _
4. First two digits: first digit times next number:
6 × 7 = 42: 4 2 _ _ _ _
5. So 654 × 656 = 429024.
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