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Prime Time Investigation 3-Factorizations: Searching for Strings of Factors
In previous Investigations, students found the factors and factor pairs of a number and the common factors and
common multiples of two or more numbers. This Investigation provides opportunities for students to think about
factorizations (a product of numbers, perhaps with some repetitions, resulting in the desired number) of whole
numbers as the products of several whole numbers. For example, two factorizations of 30 are 3 × 15 and 2 × 3 × 15.
By finding longer and longer factor strings of a number, students will find the prime factorization of a number (the
product of prime numbers, perhaps with some repetitions, resulting in the desired number). For example, the prime
factorization of 7,007 is 7 × 11 × 13. Prime factorizations CONTAIN ONLY PRIME NUMBERS and are unique to each
number (only one prime factorization for each number exists).
We end the Investigation with students using the prime factorization to find the least common multiple and greatest
common factor of two numbers, which is a much more efficient method than making lists like they were doing in
Investigation 2. Exponents are introduced as an efficient way to write repeated factors in the factorization of a number.
Prime Time Investigation 3.1-The Product Puzzle: Finding Factor Strings
In Investigation 3.1, students explore a product puzzle that includes several strings of
numbers forming factorizations of a particular number, 840 (pictured at left). The goal
is to find longer and longer strings of factorizations. Students find a factorization that
includes two or more numbers and then break each nonprime factor into a product of
factors, continuing until all the factors are prime.
For example, students may identify the factor string of 280 × 3, from there they can
break 280 into a factor pair of 140 × 2, so another factor string of 840 is 140 × 3 × 2.
They can continue to break down the numbers into factor pairs until all of the factors
are prime, which leads to the prime factorization of 840 being 2 × 2 × 2 × 3 × 5 × 7
At the end of the Problem, students recognize that prime numbers are the essential multiplicative building blocks for
whole numbers. That is, each whole number can be written as a product of primes in one and only one way.
Prime Time Investigation 3.2-Finding the Longest Factor String
In this Investigation, students continue to find the “longest” strings of factors that form a factorization. This leads to
breaking nonprime numbers into primes and recognizing that the process ends when all the numbers in the factorization
are prime. The students develop strategies for finding prime factorizations. In so doing, they will learn to use exponential
notation for repeated factors. An example of exponential notation is below.
At the end of this Problem, students will
begin to see that not only is this
factorization unique, but it can be used
to find other information about the
number, such as the GCF and LCM.
Prime Time Investigation 3.3-Using Prime Factorizations
In this Problem, students use prime factorizations of two or more numbers to find common factors and multiples. They
learn how to use prime factorization as a strategy for finding the least common multiple (LCM) and greatest common
factor (GCF) by identifying common prime factors in the factorization of each number.
Below are the strategies for finding common factors and common multiples, one strategy involves making lists while the
other involves using the prime factorization.