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0011 0010 1010 1101 0001 0100 1011 RAPID Math 1 2 4 • 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 0011 0010 1010 1101 0001 0100 1011 – 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 1 2 4 11Alternately 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. 0011 0010 1010 1101 0001 0100 1011 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. 1 12 If the number is divisible by both 3 and 4, it is also divisible by 12. 2 4 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 n5,3nn,672 divisible by 11? 0011• 0010 1010rewrite 1101 0001 the 0100digits 1011 Let’s 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. 1 2 4 Multiplication by 5, 50, 25 etc. 0011 0010 1010 1101 0001 0100 1011 • 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 1 2 4 Compute 0011 0010 1010 1101 0001 0100 1011 • • • • 68 × 5 = 87 × 5 = 452 × 5 = 27 × 50 = 1 2 4 Squaring Numbers that End in 5 0011 0010 1010 1101 0001 0100 1011 • 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. 1 2 4 • 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?? 0011 0010 1010 1101 0001 0100 1011 • 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. 1 2 4 Compute? 0011 0010 1010 1101 0001 0100 1011 • 305² = 93025 where 930 = 30×31 • So compute:• 205²= • 605²= 1 2 4 Multiplying Two Numbers Using the Difference of Two Squares 46 x 54 0011 0010 1010 1101 0001 0100 1011 1 Square the average of the two numbers Average = 50 502 = 2500 2 4 Multiplying Two Numbers Using the Difference of Two Squares 46 x 54 0011 0010 1010 1101 0001 0100 1011 1 2 Square half the difference of the two numbers 54 – 46 = 8 Half of 8 is 4 42 = 16 4 Have you ever heard of a²- b² 0011 0010 1010 1101 0001 0100 1011 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) = a2 − b2 1 2 4 This is a very handy formula for fast computation!!! Multiplying Two Numbers Using the Difference of Two Squares 0011 0010 1010 1101 0001 0100 46 1011 x 54 Subtract the two numbers to get your answer 502 – 42 = 2500 – 16 = 2484 46 x 54 = 2484 1 2 4 Practice 0011 0010 1010 1101 0001 0100 1011 36 x 44 1584 28 x 32 896 14 x 36 504 67 x 83 5561 1 2 4 To Multiply Two Numbers Ending in 5 and Differing by 10 0011 0010 1010 1101 0001 0100 1011 75 x 85 • Write down 75 1 2 • 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 4 Practice 35 x 45 1575 85 x 95 8075 65 x 75 4875 0011 0010 1010 1101 0001 0100 1011 1 2 4 Multiplying Two Numbers Squared 2 x 32 8 0011 0010 1010 1101 0001 0100 1011 • Multiply the numbers then square 8 x 3 = 24 82 x 32 = 242 = 576 1 2 4 Multiply A Number By 9 37 x 9 0011 0010 1010 1101 0001 0100 1011 Multiply the number by 10 37 x 10 = 370 1 2 Subtract the original number from the number above 370 – 37 = 363 4 Multiplying/Dividing by Factors 0011 0010 1010 1101 0001 0100 1011 Sometimes you can rapidly work a problem by multiplying/dividing by factors of the second number 144 x 15 => 144 x 3 = 432 144 x 15 = 2160 1 2 4 432 x 5 =2160 Multiplying/Dividing by Factors 0011 0010 1010 1101 0001 0100 1011 Practice problems: 16023 237 x 49 7532 296 x 28 41104 734 x 56 1 2 4 Checking Your Work By Casting Out 9’s 0011 0010 1010 1101 0001 0100 1011 To check your work by “casting out nines” you: 1 2 4 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 0011 0010 1010 1101 0001 0100 1011 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! 1 7 2 9 2 4 Checking Your Work By Casting Out 9’s 0011 0010 1010 1101 0001 0100 1011 It Gets Easier! 1 Now we get to actually “casting out nines” 2 4 When adding, leave out all nines and numbers that add to nine Checking Your Work By Casting Out 9’s 0011 0010 1010 1101 0001 0100 1011 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 1 6 2 8 2 4 Unfortunately, this only shows mistakes 8 out of 9 times, but it is still a quick check. Casting out Nines 0011 0010 1010 1101 0001 0100 1011 65324 + 89173 154497 1 2 4