For numbers less than 100, it is not very difficult to determine whether it is a prime number or not. We can also check it from the list given under` prime number’.
For testing any number greater than 100 whether it is a prime number or not, e can easily do so by the following steps.
1) We take the nearest lager integer than the appropriate square root of that number. Suppose it is `X’.
2) We test the divisibility of the given number by every prime number smaller than ‘X’.
3) If the number is not divisible by any of them, then it is a prime number.
For example 331
2
For 331, `X’ = 19, since 19 = 361.
The prime numbers less than 19 are 2, 3,5,7,11,13 and 17.
Now, try to divide 331 by these prime numbers. We see that is not completely divisible by any of these prime numbers. There fore, 331 is prime number.
Perfect number
If the sum of the divisors of `N’ excluding `N’ it self is equal to 1n’ then `N’ is called a perfect number.
For example it is 6, 28, and 496.
Foe 6, divisor are 1, 2 and 3
6: 1+2+3=6
28:1+2+4+7+14=28
496: 1+2+4+8+16+31+62+124+248=496
The sum of the reciprocals of the divisors of a perfect number including that of its own is always equal to 2.
Cyclic number: it is an integer of `n’ digits, which on being multiplied by any number from 1 through `N’, gives a product containing the same `N’ digits as in the original number and these digits are in the same cyclic order in the product.
For example 142857
2x 142857= 285714
3x 142857= 428571
5x142857= 714285
Sum of Proportion
Proportion is a very familiar and an important mathematical concept. Try to understand this. Suppose a fruit seller tells….
The mangos cost is Rupees 20/ per dozen. You want buy six fruits. Determine their cost.
As the number of mangos is half one dozen, their rate also has be half. Therefore, the price of six mangos is half Rupees 20/ i.e. Rupees 10/ in other words, you think the cost in proportion to their number.
A C
When __ = __
B D, the numbers A, B, C, D are in proportion.
When A, B, C, D are in proportion, they are respectively called first, second. Third and fourth proportional, A and D called the extremes, while B and C are called the means.
14 18
Example ___ = ____
21 27
Therefore 14, 21, 18, 27 are in proportion. 14 and 27 are the extremes while 21 and 18 are the means.
A C
Know that when ___ = ___
B D
Then the products are A x D and B x C are equal. When for numbers are in proportion, the product of the extremes is equal to the product of the means.
The concept of proportion need be restricted to only two equal ratios.
A C E
If __ = __ = ___ ……, then A, B, C, D, E, F ....are said to be in proportion.
B D F
5 20 15
Example __ = ___ = ____ so, 5, 9, 20. 36. 15, 27 are in proportion.
9 36 27