# p. 1

prime factorization and factoring using the ladder method there are several levels to the ladder · here are all the prime numbers less than 20 another example the number to factor is 52 · · · · · · · student asks will 2 go into 52 yes how many times 26 will 2 go into 26 yes how many times 13 will 2 go into 13 no will 3 go into 13 no recognize that 13 is prime so it cannot be factored any more so the prime factorization is 2 2 52 26 13 2 3 5 7 11 13 17 19 · you will begin factoring using the lowest prime number 2 then work their way through the next primes until they reach a prime number at the bottom factoring using the ladder method factoring of one number the number to factor is 24 · · · · · · ask will 2 go into 24 yes how many times 12 will 2 go into 12 yes how many times 6 will 2 go into 6 yes how many times 3 so the prime factorization is 2× 2× 2×3 or 23 × 3 notice how nicely and neatly the factors are lined up it is more organized than the usual factor tree method 2 × 2 × 13 or 22 ×13 2 2 2 24 12 6 3 -2-

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# p. 2

xmg playbook © how to factor a numerator and a denominator for simplifying fractions use the ladder to pull out the common prime factors you can use the ladder to find the greatest common factorgcf and the least common multiplelcm the fraction to simplify is · · finding the gcf and lcm using the ladder you can find the greatest common factor gcf and the least common multiple lcm gcf is 2 × 2 × 3 12 notice how these factors run along the side of the ladder and when multiplied the result is the greatest common factor lcm is 2 × 2 × 3 × 2 × 3 72 notice how these factors form the shape of an l for ladder or lcm and when multiplied the result is the least common multiple the fractions to add are · n 24 12 6 2 d 36 18 9 3 24 36 2 2 3 make a double ladder in order to factor out the common factor first label the numerator and the denominator n 2 2 3 24 12 6 d 36 18 9 ask · will 2 go into 24 and 36 yes · how many times for each · will 2 go into 12 and 18 yes · how many times for each · will 2 go into 12 and 18 no · will 3 go into 12 and 18 yes · how many times for each 3 1 2 8 6 2 3 · make a double ladder in order to factor out the least common multiple for the denominators put the denominators into the ladder ask 2 8 4 6 3 · · · · · will 2 go into 8 and 6 yes how many times for each will 2 go into 4 and 3 no will 3 go into 4 and 3 no no more factoring can be done in using the ladder you pull out the common factors leaving the simplified form numerator and denominator on the bottom rung of the ladder so 24 2 36 3 as you grow up in factoring you can take out a larger factor if you can recognize it for example on the first rung if you saw that 6 was a factor of each number you could start with that number then you would have to factor 4 and 6 on the next rung remember the factors along the outside of the ladder make the least common multiple when multiplied with each other so 2 x3 x 4 24 the lcm is 24 even better though is how this ladder method sets apart the factors for each denominator the bottom rung leaves factors which are not common between the two denominators now the student just has to multiply the two fractions by the missing factor for each denominator example 13 24 3 8 83 64 24 24 which is 11 24 3 nmcguire 2005 -4-

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