Question

why 39 is no prime number ​

Answer 1

Answer:

For 39 to be a prime number, it would have been required that 39 has only two divisors, i.e., itself and 1. However, 39 is a semiprime (also called biprime or 2-almost-prime), because it is the product of a two non-necessarily distinct prime numbers. Indeed, 39 = 3 x 13, where 3 and 13 are both prime.

Step-by-step explanation:

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Answer 2

• 39 = 13x3

This is a composite number, meaning it is a product of prime numbers.

If you are curious how to know if other numbers are prime, here is a nice tool:

Evaluate the square root of the number. For instance, here the square root of 39 is approximately 6.24.

You can be guaranteed that if 39 is composite, then you will find all prime numbers in its decomposition once you have tried to divide it by every prime less that 6.24.

Here, for example, we try the prime numbers 2, 3, and 5.

2 does not work. 3 gives us 13. This means 39 = 3x13.

5 does not work, and we can stop there because 7, the next prime, is bigger than 6.24. (This step really is trivial of course if we know our small primes well, since 3 and 13 are prime.)

Even though this seems trivial with 39, where we know quite clearly when we reach 3x13 that these are both prime and we are done, I say this because: what if instead you asked me “is 3973 prime?”

Well, now we have a procedure to determine this. The square root of 3973 is about 63.03. We have lots of ground to cover, but at least we know we need only check all prime numbers less than 63.03.

You can work your way through the primes, trying to divide 3973 by each one. I.e. divide by 2. If you get a decimal, move on to 3. Etc, etc. As soon as you hit a whole number quotient, stop.

Try this exercise and see if you can verify that for 3973, all prime numbers fail until we hit 29. Here, 3973/29 = 137. So we can now break 3973 into 29x137.

(It helps when dealing with prime numbers to memorize the first several in sequence: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, …)

We need go no further because the square root of 137 is about 11.7, and we have already searched all primes smaller than that. So, we conclude 3973 is not prime because it can be written as a product of two primes, namely, 29 and 137. (We may also conclude as a secondary result that 137 is prime.)

There are, of course, more advanced techniques to evaluating if a large number is prime, using modular arithmetic tricks, but this method is a good grassroots one for when you want to check your logic from first principles using arithmetic of the natural numbers.