Question

The prime factors of 210 are

• A. 6, 5 and 7
• B. 2, 15 and 7
• C. 2 , 3 and 35
• D. 2, 3, 5 and 7

Hint:

Prime factorization requires every listed factor to be a prime number. Numbers like 6, 15, and 35 are composite numbers, leaving Option D as the only valid prime factor set.

Answer 1

The Correct Option is D

Concept: Prime factorization involves breaking down a composite integer into a product of prime numbers. A prime number is an integer strictly greater than 1 whose only positive divisors are 1 and itself.

Step 1: Perform systematic prime factorization on the target number 210.
Since 210 is an even number ending in zero, it can be divided by the lowest prime number, which is 2: \[ 210 \div 2 = 105 \] Now, analyze the quotient 105. It is an odd number, so it is not divisible by 2. Let us check for divisibility by the next prime number, 3. The sum of the digits is \(1 + 0 + 5 = 6\), which is divisible by 3: \[ 105 \div 3 = 35 \] Now, evaluate the new quotient 35. The sum of its digits is \(3 + 5 = 8\) (not divisible by 3). Since it ends in 5, it is perfectly divisible by the next prime number, 5: \[ 35 \div 5 = 7 \] The remaining quotient is 7, which is itself a prime number: \[ 7 \div 7 = 1 \]

Step 2: Collect and write out all the prime factors.
Combining all the prime divisors used in the step-by-step reduction: \[ 210 = 2 \times 3 \times 5 \times 7 \] All elements within the set \(\{2, 3, 5, 7\}\) are prime numbers.

Step 3: Analyze the alternative options.

• In Option A: 6 is a composite number (\(2 \times 3\)), so this is not a collection of prime factors.
• In Option B: 15 is a composite number (\(3 \times 5\)), so it cannot be a prime factor.
• In Option C: 35 is a composite number (\(5 \times 7\)), so it is invalid. Therefore, Option (D) contains the complete list of prime factors.