Is 72 a prime number? What are the divisors of 72?

Is 72 a prime number?

It is possible to find out using mathematical methods whether a given integer is a prime number or not.

For 72, the answer is: No, 72 is not a prime number.

The list of all positive divisors (i.e., the list of all integers that divide 72) is as follows: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.

To be 72 a prime number, it would have been required that 72 has only two divisors, i.e., itself and 1.

As a consequence:

  • 72 is a multiple of 1
  • 72 is a multiple of 2
  • 72 is a multiple of 3
  • 72 is a multiple of 4
  • 72 is a multiple of 6
  • 72 is a multiple of 8
  • 72 is a multiple of 9
  • 72 is a multiple of 12
  • 72 is a multiple of 18
  • 72 is a multiple of 24
  • 72 is a multiple of 36

To be 72 a prime number, it would have been required that 72 has only two divisors, i.e., itself and 1.

Is 72 a deficient number?

No, 72 is not a deficient number: to be deficient, 72 should have been such that 72 is larger than the sum of its proper divisors, i.e., the divisors of 72 without 72 itself (that is 1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 18 + 24 + 36 = 123).

In fact, 72 is an abundant number; 72 is strictly smaller than the sum of its proper divisors (that is 1 + 2 + 3 + 4 + 6 + 8 + 9 + 12 + 18 + 24 + 36 = 123). The smallest abundant number is 12.

Parity of 72

72 is an even number, because it is evenly divisible by 2: 72 / 2 = 36.

Is 72 a perfect square number?

A number is a perfect square (or a square number) if its square root is an integer; that is to say, it is the product of an integer with itself. Here, the square root of 72 is about 8.485.

Thus, the square root of 72 is not an integer, and therefore 72 is not a square number.

What is the square number of 72?

The square of a number (here 72) is the result of the product of this number (72) by itself (i.e., 72 × 72); the square of 72 is sometimes called "raising 72 to the power 2", or "72 squared".

The square of 72 is 5 184 because 72 × 72 = 722 = 5 184.

As a consequence, 72 is the square root of 5 184.

Number of digits of 72

72 is a number with 2 digits.

What are the multiples of 72?

The multiples of 72 are all integers evenly divisible by 72, that is all numbers such that the remainder of the division by 72 is zero. There are infinitely many multiples of 72. The smallest multiples of 72 are:

  • 0: indeed, 0 is divisible by any natural number, and it is thus a multiple of 72 too, since 0 × 72 = 0
  • 72: indeed, 72 is a multiple of itself, since 72 is evenly divisible by 72 (we have 72 / 72 = 1, so the remainder of this division is indeed zero)
  • 144: indeed, 144 = 72 × 2
  • 216: indeed, 216 = 72 × 3
  • 288: indeed, 288 = 72 × 4
  • 360: indeed, 360 = 72 × 5
  • etc.

How to determine whether an integer is a prime number?

To determine the primality of a number, several algorithms can be used. The most naive technique is to test all divisors strictly smaller to the number of which we want to determine the primality (here 72). First, we can eliminate all even numbers greater than 2 (and hence 4, 6, 8…). Then, we can stop this check when we reach the square root of the number of which we want to determine the primality (here the square root is about 8.485). Historically, the sieve of Eratosthenes (dating from the Greek mathematics) implements this technique in a relatively efficient manner.

More modern techniques include the sieve of Atkin, probabilistic algorithms, and the cyclotomic AKS test.

Numbers near 72

  • Preceding numbers: …70, 71
  • Following numbers: 73, 74

Nearest numbers from 72

  • Preceding prime number: 71
  • Following prime number: 73
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