Divisibility
Divisibility is the ability to divide without leaving any remainder.
An integer x divides y, if there exists an integer ‘a’ such that y = ax
and thus we write as x|y (x divides y).
If x does not divides y, we write as (x does not divides y)
[This can also be stated as:
a) y is divisible by x or
b) x is a divisor of y or
c) y is a multiple of x]
Properties of Divisibility:
- 1. x|y and y|z means x|z
- 2. if x|y and x|z then x|(ay + bz) where a, b are real numbers
- 3. If x|y and y|z then x = (+/-) y
- 4. If x|y and x>0, y>o then x<y
- 5. If x|y then x|yz for any integral value of z
- 6. If x|y then nx|ny for all non zero value of n
Test of Divisibility
Divisibility by certain special numbers can be determined without actually carrying out the process of division. The following theorem summarizes the result:
A positive integer N is divisible by
i) 2 if and only if the last digit (unit’s digit) is even.
ii) 3 if and only the sum of all the digits is divisible by 3.
iii) 4 if and only if the number formed by last two digits is divisible by 4.
iv) 5 if and only if the last digit is either 0 or 5.
v) 6 if it is divisible by both 2 and 3, as per rules mentioned above
vi) 7 if
a) Adding the last digit to 3 times the rest gives a multiple of 7
b) Adding 3 times the last digit to 2 times the rest gives a multiple of 7
vii) 8 if an only if the number formed by the last three digits is divisible by 8.
viii) 9 if and only if the sum of all the digits is divisible by 9.
ix) 25 if and only if the number formed by the last two digits is divisible by 25.
x) 125 if and only if the number formed by the last three digits is divisible by 125.
xi) 11 if and only if the difference between the sum of digits in the odd places (starting from right) and sum of the digits in the even places (starting from the right) is a multiple of 11.
xii) 13 Add four times of unit’s digit for the rest, if the resultant number is divisible by 13 then the number will be divisible by 13.
Prime Numbers
In the section above, we discussed about divisibility. What if a number is not divisible by any number apart from itself and 1?
Such numbers are called Prime Numbers.
Some examples of prime numbers are 2, 3, 5, 7, 11, 13, 17, 19…..
Here are first 100 prime numbers
| Sequence | Numbers |
| 1 – 10 | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 |
| 11 – 20 | 31, 37, 41, 43, 47, 53, 59, 61, 67, 71 |
| 21 – 30 | 73, 79, 83, 89, 97, 101, 103, 107, 109, 113 |
| 31 – 40 | 127, 131, 137, 139, 149, 151, 157, 163, 167, 173 |
| 41 – 50 | 179, 181, 191, 193, 197, 199, 211, 223, 227, 229 |
| 51 – 60 | 233, 239, 241, 251, 257, 263, 269, 271, 277, 281 |
| 61 – 70 | 283, 293, 307, 311, 313, 317, 331, 337, 347, 349 |
| 71 – 80 | 353, 359, 367, 373, 379, 383, 389, 397, 401, 409 |
| 81 – 90 | 419, 421, 431, 433, 439, 443, 449, 457, 461, 463 |
| 91 – 100 | 467, 479, 487, 491, 499, 503, 509, 521, 523, 541 |
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