Question

(a)
A number is divisible by 6 if it is divisible by its two co-prime factors
and
and
(b) 43185 is divisible by 15 as it is divisible by.
(c) The number 8625 is not divisible by 6 as it is divisible by
but not by
(d) The number 54420 is divisible by 12 as it is divisible by --
- and
(e)
The number 781022 is divisible by 11 as the difference of the sum of the digits
at odd places and the sum of the digits at even places is​

Answer 1

Answer:

Step-by-step explanation:

Abstract. 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.

TABLE 1. SIMPLE DIVISIBILITY RULES FOR THE 1st 1000 PRIME NUMBERS

1. A number is divisible by 2 if the last digit of the number is divisible by 2.

2. A number is divisible by 3 if − 2 times the last digit of the number added to the rest of the number is divisible by 3.

3. A number is divisible by 5 if the last digit of the number is divisible by 5.

4. A number is divisible by 7 if − 2 times the last digit of the number added to the rest of the number is divisible by 7.

5. A number is divisible by 11 if − 1 times the last digit of the number added to the rest of the number is divisible by 11.

6. A number is divisible by 13 if − 9 times the last digit of the number added to the rest of the number is divisible by 13