Is 49 a prime number?

You surely remember number 49 for being a perfect square: $$49\;=\;7^2$$.  In fact, it is a particular one, because its digits are also perfect squares: $$4\;=\;2^2\;\;and\;9\;=\;3^2$$.

Another property of 49 is that it is a composite number, as we will study next.

We will start recalling some definitions that we have widely discussed in our Prime Numbers article. If you still feel unfamiliar with these notions, we invite you to first read that article and come back here later.

A factor of a natural number is a positive divisor of the number. A proper factor of a natural number is a factor that is different from 1 and from the number itself. For example, $$14 = 2\times 7 = 1\times 14$$; thus, 1, 2, 7, and 14 are all factors of 14, but only 2 and 7 are proper factors of 14.

A natural number is called a prime number if it is greater than 1, and it doesn’t have proper factors. For example, the first five prime numbers are 2, 3, 5, 7, and 11.

A composite number is a natural number that has proper factors. For example, 14 is a composite number because, as we just saw, it has proper factors.

Why is 49 not a prime number?

Forty-nine is not a prime number because it is a perfect square: $$49 = 7 \times 7$$. Meaning that 7 is a proper factor of 49. Therefore, 49 is a composite number.

Another way of understanding why 49 is not prime, is recalling that a prime number of objects cannot be arranged into a rectangular grid with more than one column and more than one row.

As we see below, 49 stars can certainly be arranged into a rectangular grid with seven columns and seven rows. This means that 49 is not prime!

What we can ensure is that the square root of 49 is a prime number, because $$\sqrt{49}\;=\;7$$ and 7 is prime. Also, the sum of 49’s digits is 4 + 9 = 13, which is a prime number.

Occurrences of 49 among the prime numbers

We list below some occurrences of 49 among the first few prime numbers. This way, you can have them at hand when deciding if some numbers related to 49 are primes.

First few prime numbers where 49 occurs

149, 349, 449, 491, 499

Let’s see why 149, the first on this list, is a prime number. For that, we will use the following property.

If 𝒏 is a natural number, and neither of the prime numbers less than $$\sqrt n$$ divides 𝒏, then 𝒏 is a prime number.

Notice that 149<169, thus $$\sqrt{149}\;<\;\sqrt{169}\;=\;13$$. Therefore, the prime numbers less than $$\sqrt{149}$$ are 2, 3, 5, 7 and 11. Moreover,

$$149 = (2 \times 74) + 1$$
$$149 = (3 \times 49) + 2$$
$$149 = (5 \times 29) + 4$$
$$149 = (7 \times 21) + 2$$
$$149 = (11 \times 13) + 6$$

Meaning that neither of the primes 2, 3, 5, 7, nor 11 divides 149. By the property above, we get that 149 is a prime number.

We have verified that 149 is a prime number to show the method that we just used. However, there is no need to verify that each number in the table is prime. We can assume they are, and use them when needed.

Example: Which of the numbers 249 and 349 is prime?

We first notice that 349 is in the table. Therefore, we know it is a prime number.

We also notice that 249 is not in the table. Thus, it should be a composite number. To verify that, we recall the following rule that we discussed in our Prime Numbers article.

Every prime number is of the form 6k + 1 or 6k + 5: Therefore, if a number is not of the form 6k + 1 or 6k + 5, it can’t be prime. In other words, if the remainder when a number is divided by 6 is different from 1 or 5, then the number is not prime.

Notice that 249 = 6(41) + 3. Therefore, when we divide 249 by 6, we get 3 as a remainder. This means that 249 is not of the form 6k + 1 or 6k + 5. Thus, 249 is not prime, it is composite!

Although we just verified that 249 is a composite number, we still haven’t found a proper factor for it. To do so, we divide 249 by each of the first prime numbers: 2, 3, 5, 7, etc., until we get a factor. Since 249 is not an even number, 2 doesn’t divide 249. However, 3 does divide 249, and it is, therefore, a proper factor of 249:

$$249 = 3 \times 83$$.

Now, we are more than sure that 249 is a composite number!

Do you know that 11 is a prime number?