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MrV’s Shorthand Division (by Single Digits 2, 5, and 3) It Helps You Do Prime Factorization using Factor Trees: If you are SURE that a long number divides exactly by 2, 5, or 3, you can use Shorthand Division to quickly find the other factor. Easy Tests for Divisibility: 2: Only even numbers (ending with 0, 2, 4, 6, or 8) divide by 2 Examples: 34,138 will divide by 2, but 123,243 will not 5: Only numbers ending with 0 or 5 will divide by 5 Examples: 3,005 will divide by 5, but 3,002 will not 3: Only numbers with a digit sum that divides by 3 also divide by 3 Examples: 15,201 will divide by 3, but 15,202 will not 1+5+2+0+1=9 1+5+2+0+2=10 7: There’s no easy test - Divide by 7 (use either kind of division) to see if you get a 0 remainder 4, 6, 8, and 9: Why bother? Check for the primes first. If 2 and 3 will not divide, then these other digits will not

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How to do Shorthand Division Let’s use 6408 divided by 2 as an example… Write the division with the division symbol upside down, and build the quotient underneath it --> 2 ) 6 4 0 8 Work left to right, one digit at a time 2 into 6 is 3 2 into 4 is 2 2 into 0 is 0 2 into 8 is 4 2 ) 6 4 0 8 2 ) 6 4 0 8 2 ) 6 4 0 8 2 ) 6 4 0 8 3 3 2 3 2 0 3 2 0 4 It’s easy if each little division goes evenly. But if not… ? Let’s call any remainders “carries.” (They get carried to the next digit) 2 into 7 is 3, r 1 2 into 12 is 6 2 into 1 is 0 r 1 2 into 18 is 9 2 ) 7 1 2 1 8 2 ) 7 1 2 1 8 2 ) 7 1 2 1 1 8 2 ) 7 1 2 1 1 8 3 3 6 3 6 0 3 6 0 9 Many students can do these little divisions in their heads, but some may find MrV’s Tables are useful when dividing by 3, 5 and 7 quotient

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Dividing a number by 2: Either it goes exactly, or there is a remainder of 1 You can use MrV’s Times Table. You may find it easier if you do some computations in your head Long Division: Shorthand Division: (leave some space between the digits) Long Division compared to Shorthand Division 75 2 into 8 goes 4, exactly 2 into 7 goes 3, with 1 left over – making the next digit 1 2 2 into 12 goes 6, exactly 2 into 0 goes 0, exactly 2 into 5 goes 2, with 1 left over – making the next digit 18 2 into 18 goes 9, exactly

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Dividing a number by 3: Either it goes exactly, or there is a remainder of 1 or 2 You can use MrV’s Times Table. You may find it easier if you do some computations in your head Long Division: Shorthand Division: (leave some space between the digits) Long Division compared to Shorthand Division 4 2 3 into 6 goes 2, exactly 3 into 4 goes 1, with 1 left over – making the next digit 1 2 3 into 12 goes 4, exactly 3 into 2 goes 0, with 2 left over – making the next digit 2 7 3 into 27 goes 9, exactly

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Polynomial Division with a Box. Polynomial Multiplication: Area Method x + 5 x 2 x 3 5x25x2 -4x 2 -4x-4x -20x x +1 5 Multiply (x + 5)(x 2 – 4x + 1) x.

Polynomial Division with a Box. Polynomial Multiplication: Area Method x + 5 x 2 x 3 5x25x2 -4x 2 -4x-4x -20x x +1 5 Multiply (x + 5)(x 2 – 4x + 1) x.

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