Prime numbers:

A positive integer p is considered a prime number, if p greater than 1 and p does not have factors other than 11 and p.  

Prime numbers up to 500;

    

	2	3	5	7	11	13	17	19	23
29	31	37	41	43	47	53	59	61	67
71	73	79	83	89	97	101	103	107	109
113	127	131	137	139	149	151	157	163	167
173	179	181	191	193	197	199	211	223	227
229	233	239	241	251	257	263	269	271	277
281	283	293	307	311	313	317	331	337	347
349	353	359	367	373	379	383	389	397	401
409	419	421	431	433	439	443	449	457	461
463	467	479	487	491	499

Properties of arithmetics

Reflective property A quantity is congruent (equal) to itself. a=a Symmetric property If a=b, then b=a. Transitive property If a=b and b=c, then a=c. Addition postulate If equal quantities are added to equal quantities, the sum are same Subtraction postulate If equal quantities are subtracted from equal quantities, the difference are equal. Multiplication postulate If equal quantities are multiplied by equal quantities, the products are equal. (also double of equal quantities are equal.) Division postulate If equal quantities are divided by equal none zero quantities, the quotients are equal. (also halves of equal quantities are equal.) Substitution postulate A quantity may be substituted for its equal in any expression. Partition postulate The whole is equal to some of its parts. Also between of points: AB+BC=AC. Angle addition postulate: m<ABC+ m<CBD=m<ABD

Divisibility checking

a number is a multiple of;	Then;
2	Number should be an even number (ends with 0,2,4,6,8).
3	Sum of the digits should be divisible by 3.
4	Last 2 digit of the number may be ‘00’ or the last two digits form a number, which may be divisible by 4.
5	Unit place of the number should be zero or five.
6	Sum of the digit should be divisible by 2 and 3.
8	Last three digits may be ‘000’ or the last 3 digits form a number which may be divisible by 8.
9	Sum of the digits should be divisible by 9.
10	Last digit should be zero.
11	Difference of the sum of the alternate numbers may be zero or divisible by 11.

NUMBER SYSTEM

NUMBER SYSTEM

REAL NUMBERS

 A real number is any element of the set R, which is the union of the set of rational numbers and the set of irrational numbers. in mathematical expressions, unknown or unspecified real numbers are usually represented by z.
ex: 2, 5.6,  9.75, 5½,√2,....

RATIONAL NUMBERS

A rational number is any number that can be expressed in the form of p/q, with denominator q not equal to zero. since q may be equal to 1, every integer is a rational number.

IRRATIONAL NUMBERS

An irrational number is any real number that cannot be expressed as a ratio of integers. it can not be represented as terminating or repeating decimals.ex:√2

INTEGER NUMBER

An integer is a number that can be written without a fractional component.
ex: 21, 4, 0, and −2048 are integers, while 9.75, 5½, and √2 are not.

N	Natural numbers	1, 2, 3, 4, ………
W	Whole numbers	0, 1, 2, 3, ……
Z	Integers	…….., -4, -3, -2, -1, 0, 1, 2, 3, 4, ……..
Q	Rational numbers	Any numbers in the form of  p/q ; where q not equal to zero.
	Irrational numbers	An irrational number is any real number that cannot be expressed as a ratio of integers. it cannot be represented as terminating or repeating decimals. ex:√2,
C	Complex numbers	Numbers in the form of the form a+ib

Basic operations

ARITHMETICS

Basic operations

addition
subtraction
multiplication
division

Addition and subtraction on integers:

·        If sign of the numbers match, add the numbers together and keep the  same sign.

Example 1)   -5 + (-1) = -8

                 2)   3 + 4 = 7

·        If the sign does not match, subtract the number and keep the sign of largest number.

Example: 1) -9 + 6  = -3             

                              2)  6 + (-12) = 6

·        For subtraction of two numbers, add the opposite of the number after subtraction sign.

            Example 1)   7 – 10 = 7 + (-10)   = -3      

                            2)   5 - (-3) = 5 + 3 = 8           3)   -9 - (-2) = -7

Multiplication and division:

(+ve) x (+ve) = (+ve)	(+ve) / (+ve) = (+ve)
(-ve) x (-ve) = (+ve)	(-ve) /  (-ve) = (+ve)
(-ve) x (+ve) = (-ve)	(-ve) /  (+ve) = (-ve)
(+ve) x (-ve) = (-ve)	(+ve) / (-ve) = (-ve)

Puzzle:

Einstein's Gardens Puzzle

The Puzzle:

Five friends have their gardens next to one another, where they grow three kinds of crops: fruits (apple, pear, nut, cherry), vegetables (carrot, parsley, gourd, onion) and flowers (aster, rose, tulip, lily).

 1. They grow 12 different varieties.
2. Everybody grows exactly 4 different varieties
3. Each variety is at least in one garden.
4. Only one variety is in 4 gardens.
5. Only in one garden are all 3 kinds of crops.
6. Only in one garden are all 4 varieties of one kind of crops.
7. Pear is only in the two border gardens.
8. Paul's garden is in the middle with no lily.
9. Aster grower doesn't grow vegetables.
10. Rose growers don't grow parsley.
11. Nuts grower has also gourd and parsley.
12. In the first garden are apples and cherries.
13. Only in two gardens are cherries.
14. Sam has onions and cherries.
15. Luke grows exactly two kinds of fruit.
16. Tulip is only in two gardens.
17. Apple is in a single garden.
18. Only in one garden next to Zick's is parsley.
19. Sam's garden is not on the border.
20. Hank grows neither vegetables nor asters.
21. Paul has exactly three kinds of vegetable.
Who has which garden and what is grown where?