RAPID Math Divisibility Rules 2 All even numbers

RAPID Math

• Divisibility Rules – 2 All even numbers (ending in 0, 2, 4, 6 or 8) – 3 The sum of the number’s digits is divisible by 3 – 4 The last two digits of the number form a 2 -digit number divisible by 4 – 5 The number ends in a 5 or 0 – 6 Divisible by both 2 and 3 – 7 Take the last digit, double it, and subtract it from Usually the digits that remain. Repeat until you get to a easier to divide by 7 number that you know is/is not divisible by 7. – 8 The last three digits of the number form a 3 -digit number divisible by 8 – 9 The sum of the number’s digits is divisible by 9 – 10 The number ends in a 0

11 Alternately add and subtract the digits from left to right. (You can think of the first digit as being 'added' to zero. ) If the result (including 0) is divisible by 11, the number is also. Example: to see whether 365167484 is divisible by 11, start by subtracting: [0+]3 -6+5 -1+6 -7+4 -8+4 = 0; therefore 365167484 is divisible by 11. 12 If the number is divisible by both 3 and 4, it is also divisible by 12. 13 Delete the last digit from the number, then subtract 9 times the deleted digit from the remaining number. If what is left is divisible by 13, then so is the original number.

• For what single digit value of n is the number n 5, 3 nn, 672 divisible by 11? • Let’s rewrite the digits of the number, alternating subtraction and addition signs between the digits as follows: n – 5 + 3 – n + n – 6 + 7 – 2 = n – 3. If n – 3 is divisible by 11, then the entire original number will be divisible by 11. This means we need to find a digit n, such that n – 3 is equal to 11, 22, 33, etc, and don’t forget 0!! If n = 3, then n – 3 = 0 which is divisible by 11.

Multiplication by 5, 50, 25 etc. • Halving numbers is also very easy, so rather than multiply by 5 we can put a 0 onto the number and halve it, because 5 is half of 10. So for, 44 × 5 we find half of 440 which is 220 so 44 × 5 = 220

Compute • • 68 × 5 = 87 × 5 = 452 × 5 = 27 × 50 =

Squaring Numbers that End in 5 • Squaring is multiplication in which a number is multiplied by itself: so 75 × 75 is called "75 squared" and is written 75². • The formula By One More Than the One Before provides a beautifully simple way of squaring numbers that end in 5. • Example of it, In the case of 75², we simply multiply the 7 (the number before the 5) by the next number up, 8. This gives us 56 as the first part of the answer, and the last part is simply 25 (5²).

Compute? ? • 75²= • 35²= • 65²= • 85²= Note: Example 4 Also since 4½= 4. 5, the same method applies to squaring numbers ending in ½. So 4½² = 20¼, where 20 = 4× 5 and ¼=½².

Compute? • 305² = 93025 where 930 = 30× 31 • So compute: • 205²= • 605²=

Multiplying Two Numbers Using the Difference of Two Squares 46 x 54 Square the average of the two numbers Average = 50 502 = 2500

Multiplying Two Numbers Using the Difference of Two Squares 46 x 54 Square half the difference of the two numbers 54 – 46 = 8 Half of 8 is 4 42 = 16

Have you ever heard of a²- b² WHEN THE SUM of two numbers multiplies their difference -(a + b)(a − b) -- then the product is the difference of their squares: (a + b)(a − b) = a 2 − b 2 This is a very handy formula for fast computation!!!

Multiplying Two Numbers Using the Difference of Two Squares 46 x 54 Subtract the two numbers to get your answer 502 – 42 = 2500 – 16 = 2484 46 x 54 = 2484

Practice 36 x 44 1584 28 x 32 896 14 x 36 504 67 x 83 5561

To Multiply Two Numbers Ending in 5 and Differing by 10 75 x 85 • Write down 75 • In front of the 75 write the product of the tens digit of the smaller number and the sum of the tens digit of the larger number and 1 7 x 9 = 63 75 x 85 = 6375

Practice 35 x 45 1575 85 x 95 8075 65 x 75 4875

Multiplying Two Numbers Squared 82 x 32 • Multiply the numbers then square 8 x 3 = 24 82 x 32 = 242 = 576

Multiply A Number By 9 37 x 9 Multiply the number by 10 37 x 10 = 370 Subtract the original number from the number above 370 – 37 = 363

Multiplying/Dividing by Factors Sometimes you can rapidly work a problem by multiplying/dividing by factors of the second number 144 x 15 => 144 x 3 = 432 x 5 =2160 144 x 15 = 2160

Multiplying/Dividing by Factors Practice problems: 16023 237 x 49 7532 296 x 28 41104 734 x 56

Checking Your Work By Casting Out 9’s To check your work by “casting out nines” you: First add the digits together Then keep adding the digits together till you get a one digit answer

Checking Your Work By Casting Out 9’s Example: 13579 + 24680 38259 1+3+5+7+9=25; 2+5=7 2+4+6+8+0=20; 2+0=2 3+8+2+5+9=27; 2+7=9 Thus the answer checks! BUT WAIT! 7 2 9

Checking Your Work By Casting Out 9’s It Gets Easier! Now we get to actually “casting out nines” When adding, leave out all nines and numbers that add to nine

Checking Your Work By Casting Out 9’s Example: 13579 leave out 9 and 3+7 1+5=6 + 24680 2+4+6+8+0=20; 2+0=2 38259 leave out 9 3+8+2+4=17; 7+1 6 2 8 Unfortunately, this only shows mistakes 8 out of 9 times, but it is still a quick check.

Casting out Nines 65324 + 89173 154497