Decomposing Numbers to Identify Multiples
Grade 4 Eureka Math students use the associative property of multiplication to determine whether a number is a multiple of another by decomposing its factors. If N = b × c and b is a multiple of a, then N is also a multiple of a. For instance, to check if 84 is a multiple of 6: 84 = 7 × 12, and 12 = 2 × 6, so 84 = 7 × 2 × 6 = 14 × 6, confirming 84 is a multiple of 6. This approach develops number theory fluency and connects factor decomposition to divisibility.
Key Concepts
If a number $N$ can be expressed as a product of factors, $N = b \times c$, and one of its factors, $b$, is a multiple of another number, $a$, then $N$ is also a multiple of $a$. Using the associative property, we can see that if $b = k \times a$, then: $$N = (k \times a) \times c = a \times (k \times c)$$.
Common Questions
How do you check if a number is a multiple using decomposition?
Express the number as a product of factors. If you can regroup those factors to show the target number as one of the factors, then the number is a multiple.
Is 84 a multiple of 6?
Yes. 84 = 7 × 12 = 7 × 2 × 6 = 14 × 6. Since 84 equals 6 times a whole number, it is a multiple of 6.
What property allows you to regroup factors?
The associative property of multiplication lets you change the grouping of factors without changing the product.
How does this connect to divisibility?
A number is a multiple of n if and only if it is divisible by n with no remainder. Decomposing into factors shows why.
What does this skill prepare students for?
It builds toward understanding factors, multiples, prime factorization, and divisibility rules in Grade 4 and beyond.