It is possible to find out using mathematical methods whether a given integer is a prime number or not.
Yes, 78 570 509 is a prime number.
Indeed, the definition of a prime numbers is to have exactly two distinct positive divisors, 1 and itself. A number is a divisor of another number when the remainder of Euclid’s division of the second one by the first one is zero. Concerning the number 78 570 509, the only two divisors are 1 and 78 570 509. Therefore 78 570 509 is a prime number.
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Since 78570509 is a prime number, 78570509 is also a deficient number, that is to say 78570509 is a natural integer that is strictly larger than the sum of its proper divisors, i.e., the divisors of 78570509 without 78570509 itself (that is 1, by definition!).