Write 490 as the product of its prime factors.
The correct answer and explanation is:
The prime factorization of 490 is:
490 = 2 × 5 × 7 × 7
or
490 = 2 × 5 × 7²
Explanation:
Prime factorization is the process of expressing a number as a product of its prime factors. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. The goal is to break down a composite number into the basic building blocks which are primes.
To factorize 490 into its prime factors, start by dividing by the smallest prime number possible and continue dividing the quotient until only prime numbers remain.
Step 1: Divide 490 by 2 (the smallest prime number)
490 ÷ 2 = 245
Since 2 divides 490 evenly, 2 is one prime factor.
Step 2: Now factorize 245. The smallest prime number that divides 245 is 5.
245 ÷ 5 = 49
So, 5 is another prime factor.
Step 3: Next factorize 49. It is not divisible by 2, 3, or 5. The next prime number is 7.
49 ÷ 7 = 7
So, 7 is a prime factor. Since 7 divided 49 gives 7 again, it means 7 appears twice.
Step 4: The remaining 7 is itself a prime number.
Thus, the prime factors of 490 are 2, 5, 7, and 7, or written with exponents: 2 × 5 × 7².
Prime factorization is useful in various mathematical contexts such as finding the greatest common divisor (GCD), least common multiple (LCM), simplifying fractions, and solving problems involving divisibility. It reveals the fundamental structure of a number by breaking it down to its irreducible components.