Without Using A Calculator Find The Largest Prime Factor of

What is a prime factor?

A prime factor is a prime number that divides another number exactly without leaving a remainder. For example, the prime factors of 15 are 3 and 5 because both are prime numbers and multiply to give 15 (3 × 5).

The largest prime factor of a number is simply the biggest prime number that divides it. For instance, the largest prime factor of 100 is 5 because 100 = 2 × 2 × 5 × 5.

How to find the largest prime factor

To find the largest prime factor of a number without a calculator, follow these steps:

1. Start with the smallest prime number (2) and check if it divides the number evenly.
2. If it does, divide the number by 2 and repeat the process with the quotient.
3. Continue with the next prime number (3, 5, 7, etc.) until you can't divide evenly anymore.
4. The last prime number you successfully divided by is the largest prime factor.

Step-by-step method

Step 1: Start with the smallest prime number

Begin with the smallest prime number, which is 2. Check if the number you're working with is even (divisible by 2).

Step 2: Divide by 2 until you can't anymore

If the number is even, divide it by 2 and write down the result. Continue dividing by 2 until you can't divide evenly anymore.

Step 3: Move to the next prime number

After you can't divide by 2 anymore, move to the next prime number (3) and repeat the process. Continue with 5, 7, 11, and so on.

Step 4: Identify the largest prime factor

The last prime number you successfully divided by is the largest prime factor of the original number.

Worked example

Let's find the largest prime factor of 120 using this method:

1. 120 ÷ 2 = 60 (divisible by 2)
2. 60 ÷ 2 = 30 (divisible by 2)
3. 30 ÷ 2 = 15 (divisible by 2)
4. Now we can't divide by 2 anymore, so move to the next prime number (3)
5. 15 ÷ 3 = 5 (divisible by 3)
6. Now we can't divide by 3 anymore, so move to the next prime number (5)
7. 5 ÷ 5 = 1 (divisible by 5)

The last prime number we successfully divided by is 5, so the largest prime factor of 120 is 5.

Common mistakes to avoid

• Skipping prime numbers: Always check divisibility with prime numbers in order (2, 3, 5, 7, etc.).
• Stopping too early: Continue dividing until you can't divide by the current prime number anymore.
• Missing factors: Make sure to check all possible prime factors, not just the obvious ones.
• Incorrectly identifying primes: Remember that 2 is the only even prime number.