Finding Number of Factors for 4200

To find the total number of factors of an integer, we first find its prime factorization. The number of factors can then be calculated using the exponents of the prime factors.

Step 1: Prime Factorization of 4200

We break down 4200 into its prime factors:

$4200 = 42 \times 100$

$42 = 2 \times 3 \times 7$

$100 = 10 \times 10 = (2 \times 5) \times (2 \times 5) = 2^2 \times 5^2$

Combining these:

$4200 = (2 \times 3 \times 7) \times (2^2 \times 5^2)$

$4200 = 2^{1+2} \times 3^1 \times 5^2 \times 7^1$

$4200 = 2^3 \times 3^1 \times 5^2 \times 7^1$

Step 2: Calculating the Number of Factors

If the prime factorization of a number $N$ is $p_1^{a_1} \times p_2^{a_2} \times \dots \times p_k^{a_k}$, the total number of factors (divisors) is given by the product:

$(\textit{Number of Factors}) = (a_1+1)(a_2+1)\dots(a_k+1)$

For 4200, the prime factorization is $2^3 \times 3^1 \times 5^2 \times 7^1$. The exponents are 3, 1, 2, and 1.

Applying the formula:

$(\textit{Number of Factors of } 4200) = (3+1)(1+1)(2+1)(1+1)$

$(\textit{Number of Factors of } 4200) = (4)(2)(3)(2)$

$(\textit{Number of Factors of } 4200) = 8 \times 6$

$(\textit{Number of Factors of } 4200) = 48$

Conclusion

The number of factors of 4200 is 48.