It is possible to find out using mathematical methods whether a given integer is a prime number or not.
For 137, the answer is: yes, 137 is a prime number because it has only two distinct divisors: 1 and itself (137).
As a consequence, 137 is only a multiple of 1 and 137..
Since 137 is a prime number, 137 is also a deficient number, that is to say 137 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 137 without 137 itself (that is 1, by definition!).
Parity of 137
137 is an odd number, because it is not evenly divisible by 2.
Is 137 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 137 is about 11.705.
Thus, the square root of 137 is not an integer, and therefore 137 is not a square number.
Anyway, 137 is a prime number, and a prime number cannot be a perfect square.
What is the square number of 137?
The square of a number (here 137) is the result of the product of this number (137) by itself (i.e., 137 × 137); the square of 137 is sometimes called "raising 137 to the power 2", or "137 squared".
As a consequence, 137 is the square root of 18 769.
Number of digits of 137
137 is a number with 3 digits.
What are the multiples of 137?
The multiples of 137 are all integers evenly divisible by 137, that is all numbers such that the remainder of the division by 137 is zero. There are infinitely many multiples of 137. The smallest multiples of 137 are:
- 0: indeed, 0 is divisible by any natural number, and it is thus a multiple of 137 too, since 0 × 137 = 0
- 137: indeed, 137 is a multiple of itself, since 137 is evenly divisible by 137 (we have 137 / 137 = 1, so the remainder of this division is indeed zero)
- 274: indeed, 274 = 137 × 2
- 411: indeed, 411 = 137 × 3
- 548: indeed, 548 = 137 × 4
- 685: indeed, 685 = 137 × 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 137). 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 11.705). 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.