Math  /  Numbers & Operations

QuestionNIS 5 (24-25) XL for School: Practice \& Problem Solving DUE Nov 24-11:5 3.2.PS-33 Question Help
Emma has 63 marbles, Noah has 49 marbles, and Michael has 77 marbles. Use the GCF and the Distributive Property to find the total number of marbles Emma, Noah, and Michael have.
Write each number as a product using the GCF as a factor, and apply the Distributive Property. 63+49+77=63+49+77= \square (Use the operation symbols in the math palette as needed. Do not simplify.).)

Studdy Solution

STEP 1

What is this asking? We need to find the total number of marbles Emma, Noah, and Michael have using the greatest common factor (GCF) and the Distributive Property, showing our work without simplifying. Watch out! Don't just add the numbers directly!
We *must* use the GCF and the Distributive Property as requested.

STEP 2

1. Find the GCF.
2. Apply the Distributive Property.

STEP 3

Let's break down 63\text{63} into its prime factors.
We can start by dividing by 3\text{3} since 6+3=96 + 3 = 9 is divisible by 3\text{3}. 63=32163 = 3 \cdot 21.
Now, 21\text{21} is also divisible by 3\text{3}, giving us 21=3721 = 3 \cdot 7.
So, 63=33763 = 3 \cdot 3 \cdot 7.

STEP 4

Now let's do the same for 49\text{49}.
It's 777 \cdot 7.
Nice!

STEP 5

Finally, 77\text{77}.
That's 7117 \cdot 11.
Awesome!

STEP 6

Looking at the prime factorizations: 63=33763 = 3 \cdot 3 \cdot 7, 49=7749 = 7 \cdot 7, and 77=71177 = 7 \cdot 11, the only common factor is **7**.
So, our GCF is **7**!

STEP 7

We'll rewrite each number as a product using the **GCF (7)**. 63=7963 = 7 \cdot 9 49=7749 = 7 \cdot 777=71177 = 7 \cdot 11

STEP 8

Now, we substitute these back into our original sum and apply the distributive property.
Remember, the distributive property says ab+ac=a(b+c)a \cdot b + a \cdot c = a \cdot (b + c).
In our case, 7\text{7} is like our a\text{a}.
So, we have: 63+49+77=(79)+(77)+(711)63 + 49 + 77 = (7 \cdot 9) + (7 \cdot 7) + (7 \cdot 11) Applying the distributive property, we get: 7(9+7+11)7 \cdot (9 + 7 + 11)

STEP 9

The total number of marbles, using the GCF and Distributive Property, is 7(9+7+11)7 \cdot (9 + 7 + 11).

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