Number System

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Number system

1. Basic Formulae

1.  (a + b)(a – b) = (a² – b²)
2. (a + b)² = (a² + b² + 2ab)
3. (a – b)² = (a² + b² – 2ab)
4. (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
5. (a³ + b³) = (a + b)(a² – ab + b²)
6. (a³ – b³) = (a – b)(a² + ab + b²)
7. (a³ + b³ + c³ – 3abc) = (a + b + c)(a² + b² + c² – ab – bc – ac)
8. When a + b + c = 0, then a³ + b³ + c³ = 3abc

2. Types of Numbers

I. Natural Numbers

Counting numbers 1,2,3,4,5,…1,2,3,4,5,… are called natural numbers

II. Whole Numbers

All counting numbers together with zero form the set of whole numbers.

Thus,

(i) 0 is the only whole number which is not a natural number.

(ii) Every natural number is a whole number.

III. Integers

All natural numbers, 0 and negatives of counting numbers i.e., …,−3,−2,−1,0,1,2,3,……..,−3,−2,−1,0,1,2,3,….. together form the set of integers.

(i) Positive Integers: 1,2,3,4,…..1,2,3,4,….. is the set of all positive integers.

(ii) Negative Integers: −1,−2,−3,…..−1,−2,−3,….. is the set of all negative integers.

(iii) Non-Positive and Non-Negative Integers: 0 is neither positive nor negative.

So, 0,1,2,3,….0,1,2,3,…. represents the set of non-negative integers,

while 0,−1,−2,−3,…..0,−1,−2,−3,….. represents the set of non-positive integers.

IV. Even Numbers

A number divisible by 2 is called an even number, e.g.,2,4,6,82,4,6,8, etc.

V. Odd Numbers

A number not divisible by 2 is called an odd number. e.g.,1,3,5,7,9,11,1,3,5,7,9,11, etc.

VI. Prime Numbers

A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself.

• Prime numbers up to 100 are :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.: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.
• Prime numbers Greater than 100: Let pp be a given number greater than 100. To find out whether it is prime or not, we use the following method:

Find a whole number nearly greater than the square root of pp. Let k>∗jpk>∗jp. test whether pp is divisible by any prime number less than kk. If yes, then pp is not prime. Otherwise, pp is prime. 

Example: We have to find whether 191 is a prime number or not. Now, 14>V19114>V191.

Prime numbers less than 14 are 2,3,5,7,11,13.2,3,5,7,11,13.

191 is not divisible by any of them. So, 191 is a prime number.

VII. Composite Numbers

Numbers greater than 1 which are not prime, are known as composite numbers, e.g., 4,6,8,9,10,12.4,6,8,9,10,12.

Note:

(i) 1 is neither prime nor composite.

(ii) 2 is the only even number which is prime.

(iii) There are 25 prime numbers between 1 and 100.

3. Remainder and Quotient

“The remainder is rr when pp is divided by k” means p=kq+rp=kq+r the integer qq is called the quotient.

For instance, “The remainder is 1 when 7 is divided by 3” means 7=3∗2+17=3∗2+1. Dividing both sides of p=kq+rp=kq+r by k gives the following alternative form pk=q+rkpk=q+rk

Example:

The remainder is 57 when a number is divided by 10,000. What is the remainder when the same number is divided by 1,000?

(A) 5 (B) 7 (C) 43 (D) 57 (E) 570

Solution:

Since the remainder is 57 when the number is divided by 10,000, the number can be expressed as 10,000n+5710,000n+57, where nn is an integer.

Rewriting 10,000 as 1,000∗101,000∗10 yields 10,000n+57=1,000(10n)+5710,000n+57=1,000(10n)+57

Now, since nn is an integer, 10n10n is an integer. Letting 10n=q10n=q , we get

10,000n+57=1,000∗q+5710,000n+57=1,000∗q+57

Hence, the remainder is still 57 (by the p=kq+rp=kq+r form) when the number is divided by 1,000. The answer is (D).

Method II (Alternative form):

Since the remainder is 57 when the number is divided by 10,000, the number can be expressed as 10,000n+5710,000n+57. Dividing this number by 1,000 yields

10,000n+57100010,000n+571000 =10,000n1000+571000=10,000n1000+571000 =10n+571000=10n+571000

Hence, the remainder is 57 (by the alternative form pk=q+rkpk=q+rk ), and the answer is (D).

4. Even, Odd Numbers

A number n is even if the remainder is zero when nn is divided by 2:n=2z+02:n=2z+0, or n=2zn=2z.

A number nn is odd if the remainder is one when nn is divided by 2:n=2z+12:n=2z+1.

The following properties for odd and even numbers are very useful – you should memorize them:

even * evenodd * oddeven * oddeven + evenodd + oddeven + odd=even=odd=even=even=even=oddeven * even=evenodd * odd=oddeven * odd=eveneven + even=evenodd + odd=eveneven + odd=odd

Example:

If nn is a positive integer and (n+1)(n+3)(n+1)(n+3) is odd, then (n+2)(n+4)(n+2)(n+4) must be a multiple of which one of the following?

(A) 3 (B) 5 (C) 6 (D) 8 (E) 16

Solution:

(n+1)(n+3)(n+1)(n+3) is odd only when both (n+1)(n+1) and (n+3)(n+3) are odd. This is possible only when nn is even.

Hence, n=2mn=2m, where mm is a positive integer. Then,

(n+2)(n+4)=(2m+2)(2m+4)=2(m+1)2(m+2)=4(m+1)(m+2)(n+2)(n+4)=(2m+2)(2m+4)=2(m+1)2(m+2)=4(m+1)(m+2)

=4 * (product of two consecutive positive integers, one which must be even)=4 * (product of two consecutive positive integers, one which must be even) =4 * (an even number), and this equals a number that is at least a multiple of 8=4 * (an even number), and this equals a number that is at least a multiple of 8

Hence, the answer is (D).

Questions

Level-I

Level-II

16. If both 11² and 3³ are factors of the number a * 4³ * 6² * 13¹¹, what is the smallest possible value of ‘a’?

1. 121
2. 3267
3. 363
4. 33
5. 37

17. Find the largest five digit number that is divisible by 7, 10, 15, 21 and 28.

1. 99840
2. 99900
3. 99990
4. 99960
5. 99970

18. Anita had to multiply two positive integers. Instead of taking 35 as one of the multipliers, she incorrectly took 53. As a result, the product went up by 540. What is the new product?

1. 1050
2. 540
3. 1440
4. 1520
5. 1590

Answers:

Level-I

Answer:1 Option D

Explanation:

Let the number be x.

Answer:2 Option D

Explanation:

Let the three integers be x, x + 2 and x + 4.

Then, 3x = 2(x + 4) + 3      x = 11.

 Third integer = x + 4 = 15.

Answer:3 Option B

Explanation:

Let the ten’s digit be x and unit’s digit be y.

Then, (10x + y) – (10y + x) = 36

 9(x – y) = 36

 x – y = 4.

Answer:4 Option B

Explanation:

Since the number is greater than the number obtained on reversing the digits, so the ten’s digit is greater than the unit’s digit.

Let ten’s and unit’s digits be 2x and x respectively.

Then, (10 x 2x + x) – (10x + 2x) = 36

 9x = 36

 x = 4.

 Required difference = (2x + x) – (2x – x) = 2x = 8.

Answer:5 Option B

Explanation:

 10x² + 8 + 18x = 80 + x²

 9x² + 18x – 72 = 0

 x² + 2x – 8 = 0

 (x + 4)(x – 2) = 0

 x = 2.

Answer:6 Option D

Explanation:

Let the ten’s digit be x and unit’s digit be y.

Then, x + y = 15 and x – y = 3   or   y – x = 3.

Solving x + y = 15   and   x – y = 3, we get: x = 9, y = 6.

Solving x + y = 15   and   y – x = 3, we get: x = 6, y = 9.

So, the number is either 96 or 69.

Hence, the number cannot be determined.

Answer:7 Option A

Explanation:

Let the numbers be a, b and c.

Then, a² + b² + c² = 138 and (ab + bc + ca) = 131.

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca) = 138 + 2 x 131 = 400.

 (a + b + c) = 400 = 20.

Answer:8 Option D

Explanation:

Let the ten’s digit be x and unit’s digit be y.

Then, number = 10x + y.

Number obtained by interchanging the digits = 10y + x.

 (10x + y) + (10y + x) = 11(x + y), which is divisible by 11.

Answer:9 Option A

Explanation:

Let the ten’s digit be x.

Then, unit’s digit = x + 2.

Number = 10x + (x + 2) = 11x + 2.

Sum of digits = x + (x + 2) = 2x + 2.

 (11x + 2)(2x + 2) = 144

 22x² + 26x – 140 = 0

 11x² + 13x – 70 = 0

 (x – 2)(11x + 35) = 0

 x = 2.

Hence, required number = 11x + 2 = 24.

Answer:10 Option A

Explanation:

Let the number be x.

 x² + 17x – 60 = 0

 (x + 20)(x – 3) = 0

 x = 3.

Answer:11 Option C

Explanation:

Let the numbers be x and y.

 y² = 625.

 y = 25.

 x = 15y = (15 x 25) = 375.

 Sum of the numbers = x + y = 375 + 25 = 400.

Answer:12 Option B

Explanation:

Let the numbers be x and y.

Then, xy = 120 and x² + y² = 289.

 (x + y)² = x² + y² + 2xy = 289 + (2 x 120) = 529

 x + y = 529 = 23.

Answer:13 Option B

Explanation:

Let the middle digit be x.

Then, 2x = 10 or x = 5. So, the number is either 253 or 352.

Since the number increases on reversing the digits, so the hundred’s digits is smaller than the unit’s digit.

Hence, required number = 253.

Answer:14 Option B

Explanation:

Let the numbers be x and y.

Then, x + y = 25 and x – y = 13.

4xy = (x + y)² – (x– y)²

   = (25)² – (13)²

   = (625 – 169)

   = 456

 xy = 114.

Answer:15 Option C

Explanation:

Let the numbers be x and x + 2.

Then, (x + 2)² – x² = 84

 4x + 4 = 84

 4x = 80

 x = 20.

 The required sum = x + (x + 2) = 2x + 2 = 42.

Answer:16  Option C

Explanation:

11² is a factor of the given number. 
In the given expression, a * 4³ * 6² * 13¹¹ none of the other factors, viz., 4, 6 or 13 is either a power or multiple of 11. 
Hence, “a” should include 11²
The question states that 3³ is a factor of the given number. 6² is a part of the number. 
6² can be expressed as 3² * 2². 
Therefore, if 3³ has to be a factor of the given number a * 4³ * 6² * 13¹¹, then we will need another 3 as part of the number.
Therefore, “a” should be at least 11² * 3 = 363 if the given number has to have 11² and 3³ as its factors.
Correct answer choice (C)

Answer:17  Option C

Explanation:

The number should be divisible by 10 (2, 5), 15 (3, 5), 21 (3, 7), and 28 (4, 7).

Hence, it is enough to check whether the number is divisibile by 3, 4, 5 and 7.

Test of divisibility by 3: Sum of the digits will be divisible by 3 if a number is divisible by 3.

Test of divisibility by 4: The righmost two digits viz., the units and tens digits will be divisible by 4 if a number is divisible by 4. For e.g, 1232 is divisible by 4 because 32 is divisible by 4.

Test of divisibility by 5: The units digit is either 5 or 0.

99960 is the only number which is divisible by 3, 4, 5 and 7.

Correct answer choice (C)

Answer:18  Option E

Explanation:

Let the number that Anita wanted to multiply be ‘X’.
She was expected to find the value of 35X.
Instead, she found the value of 53X. 

The difference between the value that she got (53X) and what she was expected to get (35X) is 540.

i.e., 53X – 35X = 540
or (53 – 35) * X = 540
X = 30

Therefore, the correct product = 53 * 30 = 1590

Correct answer choice (E)

,

A number system is a system of writing numbers. It is a set of symbols and rules for forming numbers from those symbols. The most common number system is the decimal number system, which uses the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 to represent numbers.

The decimal number system is a base-10 number system, which means that it uses 10 as its base. This means that each digit in a number has a value that is a multiple of 10. For example, in the number 123, the 1 is in the hundreds place, the 2 is in the tens place, and the 3 is in the ones place. The value of each digit is determined by its place value. The place value of a digit is the product of the digit and the base raised to the power of its place. For example, the place value of the 1 in 123 is 100, because 1 x 10^2 = 100.

The decimal number system is a positional number system, which means that the value of a digit depends on its position in the number. For example, the 1 in 123 is worth 100, while the 1 in 21 is worth 1.

The decimal number system is a decimal number system, which means that it is based on the number 10. This means that the number 10 is used as the base for all numbers in the system. For example, the number 100 is made up of two 10s, and the number 1000 is made up of three 10s.

The decimal number system is a place-value number system, which means that the value of a digit in a number depends on its place in the number. For example, the 1 in 123 is worth 100, while the 1 in 21 is worth 1.

The decimal number system is a positional number system, which means that the value of a digit in a number depends on its position in the number. For example, the 1 in 123 is worth 100, while the 1 in 21 is worth 1.

The decimal number system is a counting number system, which means that it is used to count things. For example, we can use the decimal number system to count the number of apples in a basket.

The decimal number system is a standard number system, which means that it is the most common number system used in the world.

The decimal number system is a universal number system, which means that it is used in all parts of the world.

The decimal number system is a natural number system, which means that it is based on the natural numbers. The natural numbers are the numbers that we use to count things, such as 1, 2, 3, and so on.

The decimal number system is a whole number system, which means that it includes all of the natural numbers, as well as the number 0.

The decimal number system is an integer number system, which means that it includes all of the whole numbers, as well as the negative whole numbers.

The decimal number system is a rational number system, which means that it includes all of the integers, as well as all of the numbers that can be expressed as a fraction of two integers.

The decimal number system is an irrational number system, which means that it includes all of the numbers that cannot be expressed as a fraction of two integers.

The decimal number system is a real number system, which means that it includes all of the rational numbers, as well as all of the irrational numbers.

The decimal number system is a complex number system, which means that it includes all of the real numbers, as well as the number i, which is the square root of -1.

Exponents are a way of writing numbers that are very large or very small. For example, the number 1000 can be written as 10^3, which means 10 multiplied by itself 3 times. The number 0.001 can be written as 10^-3, which means 1 divided by 10 multiplied by itself 3 times.

Roots are a way of finding the number that, when multiplied by itself, equals a given number. For example, the square root of 9 is 3, because 3 multiplied by itself equals 9. The cube root of 27 is 3, because 3 multiplied by itself 3 times equals 27.

Logarithms are a way of writing exponents. For example, the logarithm of

Algebra

1. What is algebra?
   Algebra is a branch of mathematics that deals with the study of mathematical symbols and their properties. It is used to solve equations, inequalities, and other mathematical problems.

2. What are the different types of algebra?
   There are many different types of algebra, including linear algebra, abstract algebra, and commutative algebra. Linear algebra is the study of linear equations and linear transformations. Abstract algebra is the study of algebraic structures, such as groups, rings, and fields. Commutative algebra is the study of commutative rings and algebras.

3. What are some common algebra formulas?
   Some common algebra formulas include the quadratic formula, the Pythagorean theorem, and the formula for the area of a circle. The quadratic formula is used to solve quadratic equations. The Pythagorean theorem is used to find the length of the hypotenuse of a right triangle. The formula for the area of a circle is $A = \pi r^2$, where $r$ is the radius of the circle.

Geometry

1. What is geometry?
   Geometry is the study of shapes and their properties. It is used to solve problems involving areas, volumes, and angles.

2. What are the different types of geometry?
   There are many different types of geometry, including Euclidean geometry, non-Euclidean geometry, and analytic geometry. Euclidean geometry is the study of shapes and their properties in a flat, two-dimensional space. Non-Euclidean geometry is the study of shapes and their properties in a curved, three-dimensional space. Analytic geometry is the study of shapes and their properties using coordinate geometry.

3. What are some common geometry formulas?
   Some common geometry formulas include the Pythagorean theorem, the formula for the area of a triangle, and the formula for the volume of a sphere. The Pythagorean theorem is used to find the length of the hypotenuse of a right triangle. The formula for the area of a triangle is $A = \frac{1}{2}bh$, where $b$ is the base of the triangle and $h$ is the height of the triangle. The formula for the volume of a sphere is $V = \frac{4}{3}\pi r^3$, where $r$ is the radius of the sphere.

Calculus

1. What is calculus?
   Calculus is the study of change. It is used to solve problems involving motion, velocity, acceleration, and other rates of change.

2. What are the different types of calculus?
   There are two main types of calculus: differential calculus and integral calculus. Differential calculus is the study of how functions change. Integral calculus is the study of how to find the area under a curve.

3. What are some common calculus formulas?
   Some common calculus formulas include the derivative of $x^n$, the integral of $x^n$, and the CHAIN RULE. The derivative of $x^n$ is $n x^{n-1}$. The integral of $x^n$ is $\frac{x^{n+1}}{n+1} + C$, where $C$ is an arbitrary constant. The chain rule is $\frac{d}{dx}[u(v(x))] = u'(v(x)) v'(x)$.

Statistics

1. What is statistics?
   Statistics is the study of the collection, organization, analysis, interpretation, presentation, and Communication of data. It is used to make inferences about a Population based on a sample.

2. What are the different types of statistics?
   There are many different types of statistics, including descriptive statistics, inferential statistics, and Probability. Descriptive statistics is the study of how to summarize data. Inferential statistics is the study of how to make inferences about a population based on a sample. Probability is the study of the likelihood of events occurring.

3. What are some common statistics formulas?
   Some common statistics formulas include the mean, the Median, and the mode. The mean is the Average of a set of numbers. The median is the middle number in a set of numbers. The mode is the most frequent number in a set of numbers.

1. What is the sum of the first 100 natural numbers?
   (A) 5050
   (B) 5051
   (C) 5052
   (D) 5053

2. What is the product of the first 100 natural numbers?
   (A) 3628800
   (B) 3628801
   (C) 3628802
   (D) 3628803

3. What is the greatest common factor of 12 and 36?
   (A) 4
   (B) 6
   (C) 12
   (D) 36

4. What is the least common multiple of 12 and 36?
   (A) 12
   (B) 24
   (C) 36
   (D) 72

5. What is the next number in the sequence 2, 4, 8, 16, 32, 64?
   (A) 128
   (B) 256
   (C) 512
   (D) 1024

6. What is the next number in the sequence 1, 1, 2, 3, 5, 8?
   (A) 13
   (B) 21
   (C) 34
   (D) 55

7. What is the next number in the sequence 2, 3, 5, 7, 11, 13?
   (A) 17
   (B) 19
   (C) 23
   (D) 29

8. What is the next number in the sequence 1, 4, 9, 16, 25, 36?
   (A) 49
   (B) 64
   (C) 81
   (D) 100

9. What is the next number in the sequence 2, 6, 12, 20, 30, 42?
   (A) 56
   (B) 66
   (C) 78
   (D) 90

10. What is the next number in the sequence 3, 6, 10, 15, 21, 28?
    (A) 36
    (B) 45
    (C) 55
    (D) 66