The number $M = 7^2 \times 23 \times 31$ consists of the prime factorization of $M$, where the prime numbers are 7, 23, and 31, and 7 is raised to the power of 2. Let's find all the factors of $M$ by using the prime factorization directly without computing the product.

Steps to find all factors:

1. Prime factorization: The prime factorization of $M$ is $7^2 \times 23^1 \times 31^1$.

2. Number of factors: The total number of factors of a number is determined by adding 1 to each of the exponents in its prime factorization and multiplying these results.

   • For $7^2$, there are $2 + 1 = 3$ possible factors: $7^0, 7^1, 7^2$.
   • For $23^1$, there are $1 + 1 = 2$ possible factors: $23^0, 23^1$.
   • For $31^1$, there are $1 + 1 = 2$ possible factors: $31^0, 31^1$.

   So, the total number of factors is: $3 \times 2 \times 2 = 12$ There are 12 factors in total.

3. Generating factors: To find all the factors, we take all combinations of powers of 7, 23, and 31:

   • Factors of the form $7^a \times 23^b \times 31^c$, where $a = 0, 1, 2$, $b = 0, 1$, and $c = 0, 1$.

   Here are all the possible combinations:

   • $a = 0, b = 0, c = 0$: $7^0 \times 23^0 \times 31^0 = 1$
   • $a = 1, b = 0, c = 0$: $7^1 \times 23^0 \times 31^0 = 7$
   • $a = 2, b = 0, c = 0$: $7^2 \times 23^0 \times 31^0 = 49$
   • $a = 0, b = 1, c = 0$: $7^0 \times 23^1 \times 31^0 = 23$
   • $a = 1, b = 1, c = 0$: $7^1 \times 23^1 \times 31^0 = 7 \times 23 = 161$
   • $a = 2, b = 1, c = 0$: $7^2 \times 23^1 \times 31^0 = 49 \times 23 = 1127$
   • $a = 0, b = 0, c = 1$: $7^0 \times 23^0 \times 31^1 = 31$
   • $a = 1, b = 0, c = 1$: $7^1 \times 23^0 \times 31^1 = 7 \times 31 = 217$
   • $a = 2, b = 0, c = 1$: $7^2 \times 23^0 \times 31^1 = 49 \times 31 = 1519$
   • $a = 0, b = 1, c = 1$: $7^0 \times 23^1 \times 31^1 = 23 \times 31 = 713$
   • $a = 1, b = 1, c = 1$: $7^1 \times 23^1 \times 31^1 = 7 \times 23 \times 31 = 4973$
   • $a = 2, b = 1, c = 1$: $7^2 \times 23^1 \times 31^1 = 49 \times 23 \times 31 = 34693$

List of all factors:

$1, 7, 23, 31, 49, 161, 217, 713, 1127, 1519, 4973, 34693$

These are all the factors of $M$, found by systematically combining the powers of the primes.

Would you like more details on how these combinations were formed, or do you have any other questions?

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Here are 5 related questions to deepen understanding:

1. How do you compute the number of divisors from a prime factorization?
2. What is the difference between finding prime factors and finding all factors of a number?
3. Can you explain how to find the greatest common divisor using prime factorization?
4. How would the process change if $7$, $23$, and $31$ were not prime numbers?
5. What are the applications of factorization in number theory or algebra?

Tip: For large numbers, prime factorization makes finding divisors and solving equations much easier.