It is possible to find out using mathematical methods whether a given integer is a prime number or not.
For 67, the answer is: yes, 67 is a prime number because it has only two distinct divisors: 1 and itself (67).
As a consequence, 67 is only a multiple of 1 and 67..
Since 67 is a prime number, 67 is also a deficient number, that is to say 67 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 67 without 67 itself (that is 1, by definition!).
Parity of 67
67 is an odd number, because it is not evenly divisible by 2.
Is 67 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 67 is about 8.185.
Thus, the square root of 67 is not an integer, and therefore 67 is not a square number.
Anyway, 67 is a prime number, and a prime number cannot be a perfect square.
What is the square number of 67?
The square of a number (here 67) is the result of the product of this number (67) by itself (i.e., 67 × 67); the square of 67 is sometimes called "raising 67 to the power 2", or "67 squared".
As a consequence, 67 is the square root of 4 489.
Number of digits of 67
67 is a number with 2 digits.
What are the multiples of 67?
The multiples of 67 are all integers evenly divisible by 67, that is all numbers such that the remainder of the division by 67 is zero. There are infinitely many multiples of 67. The smallest multiples of 67 are:
- 0: indeed, 0 is divisible by any natural number, and it is thus a multiple of 67 too, since 0 × 67 = 0
- 67: indeed, 67 is a multiple of itself, since 67 is evenly divisible by 67 (we have 67 / 67 = 1, so the remainder of this division is indeed zero)
- 134: indeed, 134 = 67 × 2
- 201: indeed, 201 = 67 × 3
- 268: indeed, 268 = 67 × 4
- 335: indeed, 335 = 67 × 5
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 67). 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 8.185). 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.