Factors
Factors of a number, divide the number completely.
If a, b, c, d __ are factors of "m" then 'm will be completely divisible by a, b, c, d__.
How to Get Factors of a Number
Factors of a number can be found by hit and trial method. Get any number, if it divides completely the number whose factor is to be found, it is a factor of that number. Let us discuss some rules of divisibility in order to easily find the factors of a number.
Rules of divisibility
(a) The numbers which have 0, 2, 4, 6, or 8 at the unit place is divisible by 2. For example: 24434, 21450, 231545452218 are divisible by 2.
(c) If sum of digits of a number is divisible by 3 then the number is divisible by 3. For example: Sum of the digits of 276 = 2 + 7 + 6 = 15. 15 is divisible by 3, therefore, 276 is divisible by 3.
(d) If the number formed by two digits from right side of a number is divisible by 4 the number is divisible by 4. For example: 28 in 5428 is divisible by 4, therefore, 5428 is divisible by 4
(e) If a number has the digit 0 or 5 at unit place, the number is divisible by 5. For example: 0 is at the unit place in the number 5450, therefore, .5450 is divisible by 5.
(f) If an even number is divisible by 3 then the number is divisible by 6. For example: 558 is an even number and divisible by 3, therefore, 558 is divisible by 6
(g) If the number formed by three digits from right side of a number is divisible by 8 then the number is divisible by 8. For example: 248 in 56248 is divisible by 8, thus 56248 is divisible by 8.
(h) If sum of digits of a number is divisible by 9, the number is divisible by 9. For example: Sum of digits of 5689485 = 5 + 6 + 8 + 9 + 4 + 8 + 5 = 45 and 45 is the divisible by 9. Thus 9689485 is divisible by 9.
(i) If a number has the digit 0 at the unit place, the number is divisible by 10. For example: 0 is at the unit place in the number 4560, 4560 is divisible by 10.
(j) If difference of the sum of the alternate digits of a number is either 0 or divisible by 11, the number is divisible by 11. For example: Difference of the sum of the alternative digits of 5478693 = (5+7+6+3)-(4+8+9) = 0 Thus 5478693 is divisible by 11.
(k) If a number has two prime factors then product of the prime factors is also a factor of the number. For more... 
Introduction
We have studied about the operations on numbers. In this chapter we will study about two important terms Factors and Multiples which are related to the operations on multiplication and division. Let's discuss about them. 
Division
Division: distribution of a quantity into some parts such that each of the parts contains equal amount is called division.
Dividend: The number which is divided. Divisor: The number by which other number is divided.
Quotient: Number of times by which the divisor is multiplied to get equal to dividend.
Remainder: The number which is left after subtraction
Divide 4575 by 20 . \[\begin{align} & \underset{\text{Divisor}\,\leftarrow 20)4575(228\to \text{Quotient}}{\mathop{\overset{\text{Dividend}}{\mathop{\uparrow }}\,}}\, \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{-40}{057} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{-40}{175} \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{-160}{\underline{015}}\to \text{Remainder} \\ \end{align}\]
Divide 25689 by 23 and find remainder.
\[\begin{align} & 23)25689(1116 \\ & \underline{-23} \\ & \,\,\,026 \\ & \underline{-23} \\ & \,\,\,038 \\ & \underline{-23} \\ & \,\,\,\,159 \\ & \underline{-138} \\ & \,\,\,\,\underline{021} \\ \end{align}\]
How to Perform Division
Consider first digit of the dividend from left.
Condition 1: When the first digit is greater than or equal to the divisor:
Step 1: Multiply the divisor by a number such that the product is closest to the number but not greater.
Step 2: Write the multiplier in the quotient side and the product below the number and subtract.
Step 3: Bring down the second digit of the dividend and write it right to the difference.
Step 4: Now compare the formed number with the divisor. If the number is greater than divisor, follow the above steps. If the number is smaller than the divisor multiply the divisor by 0 and follow above steps. Continue the process till all the digits of the dividend is brought down.
Condition 2: When the first digit is smaller than the divisor, Step 1: Multiply the divisor by 0 and follow the above steps.
Note: If the number formed by first two digits of the dividend is also smaller than divisor, multiply the divisor by 0 and follow the above steps.
Word Problems Based on Division
Arwin types 66 words in a minute, the average number of letter in a word is 6. If he types for 10 hours, how many times can he type the word "mathematics"?
(a) 21600 times
(b) 19200 times
(c) 15200 times
(d) 25200 times
(e) None of these
Answer: (a)
Explanation
Arwin types 66 words in a minute and the average number of letter in a word is 6 Therefore, total number of letters he types in a minute \[=66\times 6=396\] Thus, in 10 hours he can type \[=396\times 60\times 10\] letters = 237600 letters The word "mathematics" contains 11 letters. Therefore, total number of times he can type the word "mathematics" \[=\frac{237600}{11}=21600.\]
There are 72575 mangoes. They are more... 
Multiplication
When a number is added to itself number of times, the process of addition becomes bigger and lengthy,therefore, a short cut method of addition was developed to perform such additions called multiplication. Thus multiplication is a short cut method of repeated addition.
6035 + 6035 + 6035 + ...168 times. It is a tedious job to solve it. So a simple method is used to solve such questions. We can solve the above problem simply as 6035\[\times \]168 = 1013880
Multiplicand: The number which is multiplied is called multiplicand.
Multiplier: The number by which other number is multiplied is called multiplier.
Product: The result is called product.
Multiply 46898 and 64
Solution:
\[\frac{\frac{\begin{align} & \begin{matrix} 4 & 6 & 8 & 9 & 8 & {} \\ \end{matrix} \\ & \begin{matrix} \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\times & 6 & 4 \\ \end{matrix} \\ \end{align}}{\begin{align} & \,\,\,\,\begin{matrix} {} & 1 & 8 & 7 & 5 & 9\,\,\,\,2 \\ \end{matrix} \\ & \begin{matrix} +2 & 8 & 1 & 3 & 8 & 8 \\ \end{matrix} \\ \end{align}}}{\begin{matrix} 3 & 0 & 0 & 1 & 4 & 7 & 2 \\ \end{matrix}}\begin{matrix} \text{Multiplicand} \\ \text{multiplier} \\ \text{Produce} \\ \end{matrix}\]
How to Perform Multiplication
Multiply 890456 by 983
Step 1: Multiply 890456 by 3 (because 3 is at the unit place in the multiplier 983) \[890456\times 3=2671368\]
Step 2: Now multiply 890456 by 80 (because 8 is at the tens place in the multiplier \[983)890456\times 80=71236480\]
Step 3: Now multiply 890456 by 900 (because 9 is at the hundreds place in the multiplier \[983)890456\times 900=801410400~-1\]
Step 4: Now add all the above results 2671368 + 71236480 + 801410400 = 875318248
Thus \[890456\times 983=875318248\]
It can be written as
\[\begin{align} & \frac{\begin{align} & 890456 \\ & \,\,\,\,\,\times 983 \\ \end{align}}{\,\,\,\,\,\,2671368} \\ & +71236480 \\ & \frac{801410400}{875318248} \\ \end{align}\]
Find the product of 4568 and 305
Solution:
\[\frac{\frac{\begin{matrix} \begin{matrix} 4 & 5 \\ \times & 3 \\ \end{matrix}\,\,\,\,\,\,\,\,\,\begin{matrix} 6 & 8 \\ 0 & 5 \\ \end{matrix} \\ \end{matrix}}{\begin{align} & \,\,\,\,\,\,\,\,\,\begin{matrix} 2 & 2 & 8 & 4 & 0 \\ \end{matrix} \\ & \begin{matrix} + & 0 & 0 & 0 & 0\,\,\,0 \\ \end{matrix} \\ & \begin{matrix} 1 & 3 & 7 & 0 & 4 \\ \end{matrix} \\ \end{align}}}{\begin{matrix} 1 & 3 & 9 & 3 & 2 & 4 & 0 \\ \end{matrix}}\begin{matrix} \text{Multiplicand} \\ \text{multiplier} \\ \text{Product} \\ \end{matrix}\]
Word Problems Based on Multiplication
There are 225 godowns. Each godowns has 5478 rice bags and more... 
Subtraction
Subtraction: Under the operation of subtraction, difference between two numbersis found.
Minuend: The number from which other number is subtracted is called minuend.
Subtrahend: The number which is subtracted from a number is called subtrahend.
Difference: The result we get after subtraction is called difference.
Subtract 1254 from 7889
Solution:
| Thousands | Hundreds | Tens | Ones | |
| 7 | 8 | 8 | 9 | Minuend |
| -1 | 2 | 5 | 4 | Subtrahend |
| 6 | 6 | 3 | 5 | Difference |
How to Perform Subtraction
Just like addition digits of both minuend and subtrahend are arranged in the place value chart. Digits of the subtrahend comes below to the digits of minuend. Now subtract ones from ones, tens from tens and so on.
Find the difference between 4568 and 2795
Solution:
| Thousands | Hundreds | Tens | more...
 Addition
Under the operation of addition two or more than two numbers are added with each other.
Addends: The numbers that are added together are called addends.
Sum: The result we get after addition is called sum.
Add 2134 and 3425
Solution:
 How to Perform Addition
Arrange the digits of the numbers that are to be added column wise in the place value chart such that digits at the unit place in the numbers come in the units column digits at the tens place come in the tens column and so on. Now add ones to ones, tens to tens and so on.
Find the sum of 46898 and 2564
Ten thousands Thousands Hundreds Tens Ones
4 6 8 9 8
+ 2 5 6 4
____________________________________________________________________________________
4 9 4 6 2
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Word Problems Based on Addition
Population of a small town was 45789856 in the year 2009, In 2010, 214500 more people come in the town from outside and settled. Find the population of the city in 2010.
Solutions:
Population of the city in 2009 = 45789856
Number of people who come into the town = 214500
Therefore, population of the city in 2010 = 45789856 + 214500 =46004356
A factory produces 894500 cars, 789600 bikes, and 456800 scooty more...  Introduction
In the previous chapter, we have studied about numbers, way of numeration and some properties of numbers. In this chapter we will study about operation on numbers. Addition, subtraction, multiplication and division are four basic arithmetic operations. Let us know about them.  Rules for Using Symbols
Rule 1: When a symbol is repeated, its value is multiplied as many times as the symbol repeated.
II = 1 x 2 = 2,
III = 1 x 3 = 3,
XX = 10 x 2 = 20,
XXX = 10 x 3 = 30.
Note: A symbol cannot be repeated more than 3 times.
50 cannot be written as XXXXX.
Rule 2: The symbols whose value are power of 10 can be repeated. In other word the symbols, I, X, C, M can be repeated.
XXX = 30 , MM = 2000
Rule 3: The symbols whose value are either 5 or product of 5 and a power of 10 cannot be repeated. In other words V, L, and D cannot be repeated.
It is wrong to write LL for 100.
Rule 4: If a symbol of smaller value is right to the symbol of greater value, their values are added.
VI = 5 + 1 = 6, XV = 15, LX = 50 + 10 = 60.
Rule 5: If a symbol of smaller value is left to the symbol of greater value, their difference is the resulting value.
IV = 5 - 1 = 4, IX = 10 - 1 = 9.
Rule 6: The symbols whose value are either 5 or product of 5 and a power of 10 never subtracted. In other word V, L, and D are never written left to the symbol of greater value.
V cannot be written immediately left to X, L, C, D, and M; L can be written immediately left to the C, D, and M; D cannot be written left to the M.
Rule 7: If a symbol of smaller value comes between two symbols of larger value, its value is subtracted from the value of the symbol which is right to it.
XIX = 10+(10 - 1) = 19, L1V = 50 + (5 - 1).
Rule 8: A symbol of smaller value cannot be repeated two or more than two times left to a symbol of larger value.
30 cannot be written as XXL, similarly 8 cannot be written as IIX.
Rule 9: When a bar (horizontal line) is placed over a Roman symbol, the value of the symbol is increased by 1000 times.
\[\overline{\mathbf{I}}=\mathbf{1000},\text{ }\overline{\mathbf{v}}=\mathbf{5000}.\]
more...  Introduction
In this chapter, we will describe a primitive system of numeration i.e. Roman system of numeration or Roman Numerals". Romans had its own method to represent the numbers. Let us understand the way of Roman Numeration.  System of Numeration
Mathematical notation of numbers is called numeration. We are aware of the symbols which are used to write any numbers. In this chapter we will study about the following two system of numeration:
(a) Indian system of numeration
(b) International system of numeration
 Indian System of Numeration
Indian system of numeration is also called Hindu-Arabic number system. It is a positional decimal number system. Look at the following place value chart:
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