Write 42 as a product of prime factors.
42 =
To express the integer \(42\) as a product of its prime factors, we perform a prime factorization. This involves systematically dividing the number by the smallest possible prime numbers until we reach 1. A prime number is a natural number greater than 1 that is only divisible by 1 and itself. The first few prime numbers are \(2, 3, 5, 7, 11, \dots\). We aim to decompose \(42\) into these prime components.
Step 1: Check if \(42\) is divisible by the smallest prime number, \(2\). Since \(42\) is an even number, it is divisible by \(2\):
\[42 \div 2 = 21\]
So, we can write \(42 = 2 \times 21\).
Step 2: Now consider the quotient \(21\). Check if \(21\) is divisible by the smallest prime number \(2\). It is not, as it is an odd number. Moving to the next prime number, \(3\), we find:
\[21 \div 3 = 7\]
So, we can write \(21 = 3 \times 7\). Substituting this back into our expression for \(42\), we get \(42 = 2 \times 3 \times 7\).
Step 3: Finally, we examine the resulting quotient \(7\). Since \(7\) is a prime number itself, it cannot be factored any further. Thus, the prime factorization of \(42\) is complete and is given by the product \(2 \times 3 \times 7\).
The product of the prime factors of \(42\) is \(2 \times 3 \times 7\).
\[42 = \boxed{2 \times 3 \times 7}\]
Prime factorization is used in cryptography, where large numbers are factored to create secure keys. On a simpler scale, it helps in finding the greatest common divisor or least common multiple of numbers, which is essential for simplifying fractions or finding common intervals in scheduling events. For example, if you have two gears with 42 teeth and another with a different number, knowing the prime factors \(2, 3, 7\) helps you determine how they interact and mesh together efficiently.