Answer:
To list some factors:
[tex]\text{For } a = 0, b = 0, c = 0: \quad 1[/tex]
[tex]\text{For } a = 1, b = 0, c = 0: \quad 2[/tex]
[tex] \text{For } a = 0, b = 1, c = 0: \quad 3[/tex]
[tex]\text{For } a = 0, b = 0, c = 1: \quad 13[/tex]
Therefore, 8424 has 40 factors, including but not limited to 1, 2, 3, 4, 6, 8, 9, 12, 13, 18, 24, 26, 27, 36, 39, 52, 54, 78, 104, 108, 117, 156, 162, 208, 234, 312, 324, 412, 468, 624, 702, 936, 1026, 1404, 1872, 2106, 2808, 4212, and 8424.
Step-by-step explanation:
To determine the factors of the product ( 324 x 26 ), let's first calculate the product:
[tex]324 \times 26 = 8424[/tex]
Now, we need to find the factors of 8424. A systematic approach involves prime factorization.
Step 1: Prime Factorization of 324
First, break down 324:
[tex]\[ 324 \div 2 = 162 \][/tex]
[tex]\[ 162 \div 2 = 81 \][/tex]
[tex]\[ 81 \div 3 = 27 \][/tex]
[tex]\[ 27 \div 3 = 9 \][/tex]
[tex]\[ 9 \div 3 = 3 \][/tex]
[tex]\[ 3 \div 3 = 1 \][/tex]
The prime factorization of 324 is:
[tex]324 = 2^2 \times 3^4[/tex]
Step 2: Prime Factorization of 26
Next, break down 26:
[tex]26 \div 2 = 13[/tex]
[tex]13 \div 13 = 1[/tex]
The prime factorization of 26 is:
[tex]26 = 2 \times 13[/tex]
Step 3: Combine the Prime Factors
Combining all the prime factors, we get:
[tex]{8424 = 324 \times 26 = (2^2 \times 3^4) \times (2 \times 13)}[/tex]
[tex]8424 = 2^3 \times 3^4 \times 13[/tex]
Step 4: Determine the Factors
To find all factors, we'll consider every combination of the exponents of the prime factors:
The factors of ( 8424 ) are generated by varying the exponents of ( 2³ ), \( 3³), and ( 13 ):
[tex]8424 = 2^a \times 3^b \times 13^c[/tex]
[tex] {\text{where} \: \( a \in \{0, 1, 2, 3\} \), \( b \in \{0, 1, 2, 3, 4\} \), and \: \( c \in \{0, 1\} \).}[/tex]
For each combination of ( a ), ( b ), and ( c ), we get a unique factor. This results in:
[tex]{(3+1) \times (4+1) \times (1+1) = 4 \times 5 \times 2 = 40 \text{ factors}}[/tex]
