SOLUTION: What are thge prime factors of 420?

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Question 125026: What are thge prime factors of 420?
Found 2 solutions by MathLover1, solver91311:
Answer by MathLover1(20850)   (Show Source): You can put this solution on YOUR website!

Solved by pluggable solver: Find factors of any number
420 is NOT a prime number: 420 = 2 * 2 * 3 * 5 * 7

Work Shown

420 is divisible by 2: 420 = 210 * 2.
210 is divisible by 2: 210 = 105 * 2.
105 is divisible by 3: 105 = 35 * 3.
35 is divisible by 5: 35 = 7 * 5.
7 is not divisible by anything.
This calculation used this Prime Factorization Algorithm.
Answer by solver91311(24713)   (Show Source): You can put this solution on YOUR website!
Step 1 is to find the integer part of the square root of your number. The square root of 420 is a little larger than 20, so the first prime number smaller than 20 becomes your limit divisor, that would be 19.

Start with 2 because that is the first prime number.

. Divides evenly, so 2 is a factor
. Divides evenly, so another 2 is a factor
. Does not divide evenly, so we have found all of the 2s

Next prime number after 2 is 3.

. Divides evenly, so 3 is a factor
. Does not divide evenly, so we have found all the 3s

Next prime number after 3 is 5

. Divides evenly, so 5 is a factor, and we can see that the 7 we have left over is prime, so that is a factor also.

Putting it all together, we have . Done. The reason we developed the limit divisor of 19 as the first step was in case we couldn't find some even divisors in the first few prime number divisors, we needed a place to stop. In this case, had we needed to go on beyond 7 as a divisor, and we had reached 19 as a divisor, we would have known that the last dividend in the process was prime. For example, let's find the prime factors of 421.

421 is odd, so 2 is not a divisor.
The sum of the digits of 421 is not divisible by 3, so 3 is not a divisor.
421 does not end in a 5 so 5 is not a divisor.
Since 7 is a factor of 420, dividing 421 by 7 would leave a remainder of 1 (this argument could be made for all of the primes less than 7 as well).
The calculator says that 11 is not an even divisor of 421.
The calculator says that 13 is not an even divisor of 421.
The calculator says that 17 is not an even divisor of 421.
The calculator says that 19 is not an even divisor of 421.

Since the next prime number after 19, namely 23, is larger than the square root of 421, 23 and any other prime larger than 23 are eliminated from consideration because if any of these numbers are factors, the other factor would have to be less than or equal to 19 and we have eliminated all of those possibilities.

Therefore, we can say definitely that 421 is prime.
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