Question

Which of the following is NOT a prime number?
a 1001 b 1301 c 1601 d 1901

Answer 1

Simple divisibility rules are given for the 1st 1000 prime numbers.
PACS numbers: 02.10.Lh
This paper presents simple divisibility rules for the 1st 1000 prime
numbers.
With two exceptions, the rules in question are based on the observation
that if M is an integer not divisible by 2 or by 5, then M divides an
arbitrary integer N if it divides
N′ = N − N mod 10
10 + (m + A × M) × (N mod 10) (1)
= N − M × (m′ − 10 × A) × (N mod 10)
10 ,
where
m = 1 − m′ × M
10 , (2)
A is any integer (positive, negative, or 0), and, for M > 0,
m′ = [3 × (M mod 10) − 2 × M] mod 16, (3)
the integer A in Eq. (1) making it possible to customize the rule.
The rules for divisibility by 2 and by 5—the two aforesaid exceptions—
are based on the observation that, for p an arbitrary non-negative integer,
2p (and, similarly, 5p) divides N if it divides N mod 10p
, since 10p
divides N mod 10p + 1 − N mod 10p
, while 2p
(and, similarly, 5p
) divides
10p = (2 × 5)p = 2p × 5p.
Simple divisibility rules for the 1st 1000 prime numbers based on the
preceding observations (with A = 0 and p = 1) appear in Table 1 (see
below), where—technically speaking—“number” means “integer,”
“times” means “multiplied by,” and, for N being “the number,” “the last
digit of the number” is given by N mod 10 and “the rest of the number” is
given by (N − N mod 10)/10.