I. Using algebraic identities

A.	5 × 17 + 5 × 3 = _______	Use the distributive property to turn it into 5 × 20 = 100.
B.	12² – 8² = ________	Use the difference of squares to think of (12 + 8)(12 – 8) = (20)(4) = 80.

II. Multiplication shortcuts

Multiply by powers of 10. These tricks are usually taught in regular math classes at the middle school level and involve annexing the right number of zeros or shifting a decimal point. They might possibly need teaching to fourth, fifth or sixth graders.
Multiply by 50. Multiply by 100 and divide by 2 (or vice-versa). Or multiply by 5 and affix a 0 – whichever is easier.
Multiply by 25. Multiply by 100 and divide by 4 (or vice-versa).
Multiply by 75. Multiply by 100, divide by 4, and multiply by 3.
Multiply by 33a. Multiply by 100 and divide by 3.
Squaring a two-digit number ending in 5. The product ends in 25. Then ly the tens digit by 1 higher and write that product in front. Example: 75² =
(Write down 25, then multiply 7 × 8.)
Multiply two 2-digit numbers in the same decade whose units digits add to 10. (E.g. 24 × 26, 87 × 83). Multiply the units digits and write that down; if the product is less than 10, make it a two-digit number by writing a 0 in front; e.g. 1 × 9 would be written as 09. Then multiply the tens digit by one higher and write that product down in front. Example:
1. 87 × 83 = 21 (7 × 3 = 21)
2. 87 × 83 = 7221 (8 × 9 = 72)
2-digit × 2-digit in general. Use FOIL in reverse from algebra. Carry digits in your head. Example: 42 × 37 = 1554.
1. 42 × 37 = 4 2 × 7 = 14; write down the 4 and carry the 1.
2. 42 × 37 = 54 4 × 7 + 2 × 3 + 1 (carried from the last step) = 35. Write down the 5 and carry the 3 in your head.
3. 42 × 37 = 1554 4 × 3 + the carried 3 = 15.
Multiply by 11. Start with writing down the ones digit of the other factor. Then add each digit to its neighbor to the right, carrying if necessary. Then write down the first digit (plus anything that is carried mentally). Example: 11 × 6823.
11 × 6823 = 3

	(Write down the 3.)
11 × 6823 = 53	(2 + 3 = 5)
11 × 6823 = 053	(8 + 2 = 10. Carry the 1 in your head.)
11 × 6823 = 5053	(6 + 8 + 1[carried] = 15. Carry the 1 again.)

11 × 6823 = 75,053 (The first digit is 6, plus the 1 we carried.)
Multiply by 12. This is sometimes referred to as “double the digit and add to its neighbor on the right.” Example: 57 × 12 = 684
1. 57 × 12 = 4 Think 7 × 2 = 14; carry the 1.
2. 57 × 12 = 84 Think 5 × 2 = 10; add 7 so 10 + 7 = 17; add the carried 1 to get 18. Write 8 and carry the 1.
3. 57 × 12 = 684 Take the 5, then add the carried 1; write 6. K. Using Difference of Squares to help in multiplying. We can use the algebra fact (a + b)(a – b) = a² – b² to multiply two numbers that are an equal distance from each other. Example: 36 × 44. Both 36 and 44 are 4 units away from 40, so think of this as (40 + 4)(40 – 4) and work it as 40² – 4² = 1600 – 16 = 1584.
Multiply by 125. Multiply by 500 and divide by 4. So, for example, 48 × 125 would be = 12× 500 = 6000.
Double and Half. For some multiplication problems (see below), it may be easier to multiply twice one factor by half the other. This is especially good for
1. An even number times a multiple of 5 (e.g. 35 × 18 = 70 × 9)
2. A number times a multiple of 11 (e.g. 22 × 13 = 11 × 26)
3. One factor that is a mixed number ending in ½. (7½ × 16 = 15 × 8)
Multiplying by 9. Multiply the other number by 10, then subtract it. (In other words, 9n = 10n – n.) Examples:

9 × 23 = 230 – 23 = 207

19 × 42 = 20 × 42 – 42 = 840 – 42 = 738

III. Sum of a series

Example: 4 + 6 + 8 + 10 + 12 = _______

Add numbers in pairs from opposite ends inwards:

(4 + 12) + (6 + 10) + 8 = 16 + 16 + 8 = 40

Example: 1 + 2 + 3 + 4 + 5 + 6 + 7 = _________

Use the formula that 1 + 2 + 3 + ... + n = , where n is the last term

(and the number of terms). So the sum is 7 × 8 ÷ 2 = 28.

In an arithmetic sequence not starting with 1 and increasing by 1, use the formula , where n is the number of terms.

Example: 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 = ____________ 10 + 24 = 34; there are 8 terms; 8 ÷ 2 = 4; so 34 × 4 = 136.

IV. Number Theory and Related Topics

Useful fact: GCF(a, b) × LCM(a, b) = a × b. (Find the GCF and LCM by whatever method seems easiest to do mentally.)
How many positive integer divisors does a whole number have?
1. Find the prime factorization of the number (use exponents).
2. Add 1 to each exponent.
3. Multiply the new exponents together; this gives the number of positive integer factors.

Example: How many positive integer divisors does 40 have? 40 = 2³ · 5¹. The number of divisors, then, is 4 × 2 = 8.

Note on adding and subtracting fractions: For mental arithmetic, it may

often be easier to use ) and then

write in lowest terms rather than changing to common denominators and

computing.

Writing a repeating decimal as a fraction:
1. If a repeating decimal has all digits after the decimal as repetends, write a fraction with the repetend as the numerator and a denominator with the same number of 9s as the number of repeating digits. Then write in lowest terms. So .
2. Repeating decimals such as :
  1. Take 58 – 5 = 53
  2. Divide by 90 (and simplify if possible):

Another example:

(Think of where a and b are digits.)

Sets

A set with n elements has 2ⁿ subsets.

VI. Harder or More Obscure Multiplication Tricks

Multiply two 3-digit numbers. Let’s write it as abc × def, where the letter represent digits.
1. Multiply ones (f × c) and write down the ones; carry.
2. Do (b × f) + (c × e) + carried digit; write down the ones and carry.
3. Do (a × f) + (c × d) + (b × e) + carried digit; write down 1s and carry.
4. Do (a × e) + (b × d) + carried digit; write down ones and carry.
5. Multiply a × d and add carried digit; write it down.
6. Example: 123 × 456
7. 6 × 3 = 18

	8
(6 × 2) + (5 × 3) + 1 = 28	88
(6 × 1) + (4 × 3) + (5 × 2) + 2 = 30	088
(5 × 1) + (4 × 2) + 3 = 16	6088
(4 × 1) + 1 = 5	56,088

Multiply a 2-digit number by 101. Write the 2-digit number next to itself.

Example: 52 × 101 = 5252.

C. Multiply a 2-digit number by 111. Example: 111 × 58

Write down the ones digit.	111 × 58 = 8
Add the tens and ones; carry.	111 × 58 = 38 (5 + 8 = 13)
Repeat the previous step.	111 × 58 = 438 (5 + 8 + 1 = 14)
Write the tens digit (and carry).	111 × 58 = 6438 (5 + 1 = 6)

Multiply a 3-digit number by 111. Use right-to-left “sweeps” of 1, 2, 3, 2,

and 1 digit. Example: 111 × 234
111 × 234 = 4	Write down the 4.
111 × 234 = 74	3 + 4 = 7
111 × 234 = 974	2 + 3 + 4 = 9
111 × 234 = 5974	2 + 3 = 5
111 × 234 = 25,974	Write down the 2.

As before, you may have to carry a 1 or 2 in your head.

Multiply a multiple of 7 by 715. Take the multiple of 7, divide it by 7, then multiply by 5. Write that result as a 3-digit number (padding with 0s if necessary), and then write it again to the left.

Example: 42 × 715. 42 ÷ 7 = 6, 6 × 5 = 30, so the answer is 30,030.

Multiply a multiple of 7 by 429. Divide the other number by 7 and multiply by 3. Write the result as a 3-digit number, then repeat the result

in front. (This works because .) Example: 56 × 429 = ? . 56

÷ 7 × 3 = 24. So the product is 24,024.

Multiply a multiple of 7 by 572. Same idea as for 429, but divide by 7 and multiply by 4. So 14 × 572 = 8008.
Multiply a multiple of 7 by 858. Same as for 429, but divide by 7 and multiply by 6. So 21 × 858 = 18,018.
Addition using compensation. Add a number to one addend and subtract it from the other to simplify the addition. For example, change 67 + 58 into 70 + 55 so that one addend is a multiple of 10.
Subtraction using compensation: Add (or subtract) the same number to minuend and subtrahend to get easier numbers to subtract. Example: