MATH: aptitude
unit digit for number raised to any power
"*(Any even number)⁴ⁿ = ...6* It means that any even number raised to any power, which is a multiple of 4, will give 6 as the units digit . *(Any odd number)⁴ⁿ = ...1* . Exception: *0, 1, 5, 6* [These are independent of power, and units digit will be the same.]
If a car covers some journey from A to B at u km/hr, and the return trip at v km/hr, then the avg speed avg of three different speed
2uv/(u+v) km/hr 3uvw/(uv + vw + uw)
Lovely goes to Patna from New Delhi at a speed of 40 km/h and returns with a speed of 60 km/h. What is her average speed during the whole journey?
48
graphical method for squaring
51²=50²+50+51=2500+101=2601 best for unit digit in 1
squaring for unit digit 5
55*55 (5+1)*5 || 25 =3025
Sum of divisors
where a, b, and c are prime number
What is x% of y?
x× y ÷ 100 It can also be seen that *x% of y = y% of x*.
prime numbers
All the numbers that are *divisible by 1 and itself only* are known as prime numbers. As mentioned earlier, primes can be *natural numbers only*. In other words, we can say that all the numbers that have *only two factors* are known as prime numbers.
Subclasses of the integers
Even and odd numbers Prime numbers Fibonacci numbers and perfect numbers
percentage and per cent
"read as x per cent or 20% is read as 20 per cent, not 20 percentage. The word *'percentage' is used whenever% is not attached* to any value.
Base System (base 10-9,8,7,6 & 7-10)
(74)₁₀ = 8 × 9¹ + 2 × 9⁰ = (82)₉ (74)₁₀ = 1 × 8² + 1 × 8¹ + 2 × 8⁰ = (112)₈ (74)₁₀ = 1 × 7² + 3 × 7¹ + 4 × 7⁰ = (134)₇ (74)₁₀ = 2 × 6² + 0 × 6¹ + 2 × 6⁰ = (202)₆ (356)7 = 3 × 72 + 5 × 71 + 6 × 70 = (188)10
last two digits for number raised to any power
(i) *(Any even number)²⁰ⁿ will give 76* However, if units digit = 0, then it will give '00' as the last two digits. (ii) *(Any odd number)²⁰ⁿ will give 01* as its last two digits (where N is any natural number). However, if units digit = 5, then it will give '25' as the last two digits.
Average of 1st n consecutive natural numbers
(n+1)/2
CLASSIFICATION OF NUMBERS/ INTEGERS
*Natural numbers* are *counting numbers*, that is, the numbers that we use to count any number of things. For example, 1, 2, 3, .... The lowest natural number is 1. *Whole Numbers* When *zero is included* in the list of natural numbers, then they are known as whole numbers. For example, 0, 1, 2, ... The lowest whole number is 0. *Integers* are whole numbers, negative of whole numbers, and zero. *rational number* is a number that can be expressed as a *fraction* with an integer numerator and a positive integer denominator *Real numbers* are usually represented by using *decimal numerals*, in which a decimal point is placed to the right of the digit with place value 1. *complex number* is a number that can be expressed in the form *a + bi*, where a and b are real numbers and i is the imaginary unit, satisfying the equation i²= −1.
Simple Interest and Compound Interest : Remember
1 If the rate of interest = R% per annum for both CI and SI, then the difference between CI and SI for 2 yr will be equal to (R% of R)% of principal = *(R²/100) % of principal*. 2. In the above case, R = 10%, so the di erence between CI and SI for 2 yr is 1%. 2. If a sum doubles itself in n years at SI, then rate of interest = *100/n* 3. At SI, if a sum of money amount to n times in t years, then rate of interest = *(n-1)/T ×100%*
Squaring Properties
1. n² equal to the sum of the first n odd numbers *n²=∑(2k-1)* 5² = 25 = 1 + 3 + 5 + 7 + 9. square of any number can be represented as the sum 1 + 1 + 2 + 2 +...+ n − 1 + n − 1 + n. This is the result of *adding a column and row of thickness 1 to the square graph* * 52 = 50² + 50 + 51 + 51 + 52 = 2500 + 204 = 2704*. 2. only end with digits *00, 1, 4, 6, 9, or 25* in base 10, as follows: 3. if the last digit of a number is 0, its square ends in 0 (00); preceding digits form *square* 1 or 9, its square ends in 1; preceding digit *÷4* 2 or 8, its square ends in 4; preceding digit *even* 3 or 7, its square ends in 9; preceding digit *÷4* 4 or 6, its square ends in 6; preceding digit *odd* 5, its square ends in 25 ;preceding digit 0, 2, 06, 56 4. The digital sum of any perfect square can be only *0, 1, 4, 9, and 7*
Base Method Multiplication
105×107 100+(5+7)|35 (5×7) =11235 97×102 =100-1 | -06 =9900-06 =9894 better to use for base 10,100,..
mean method for squaring
52² 54×50+2²=2704 (x+y)(x-y)=x²-y² make (x-y) ends with 0
Composite Numbers
A number is composite if it is the *product of two or more than two distinct or same prime numbers*. For example, 4, 6, 8,.... 4 = 2² 6 = 2¹ × 3¹ The lowest composite number is 4. All the composite numbers will have at *least 3 factors*.
LCM × HCF
= product of two numbers. This formula can be applied only in the case of two numbers. However, if the numbers are relatively prime to each other (i.e., HCF of numbers = 1), then this formula can be applied for any number of numbers.
Percentage change
=(Change/ Initial value) ×100
Odd number
Any *odd number raised to a positive integer* power will also be an odd number, and therefore, if x is an odd number, then x², x³, x⁴, etc., will be odd numbers. .*square of odd number are odd* .square roots of odd square numbers are odd. The concept of even and odd numbers are most easily understood in the *binary base*. The above mentioned definition simply states that even numbers end with a 0 and *odd numbers end with a 1*.
Converting (101110010)2 to Octal ()8 system: Converting (101110010)2 to hexa-decimal ()16 system:
At first, we will club three digits of binary number into a single block, and then, we will write the decimal equivalent of each group (left to right). Therefore, (101110010)2 is now (101)2 (110)2 (010)2. Now, (101)2 = 1 × 22 + 0 + 1 × 20 = 5 (110)2 = 1 × 22 + 1 × 21 + 0 × 20 = 6 (010)2 = 0 × 22 + 1 × 21 + 0 × 20 = 2. Thus, (101110010)2 = (562)8 Converting (101110010)2 to hexa-decimal ()16 system: At first, we will club four digits of binary number into a single block, and then, we will write the decimal equivalent of each group (left to right). Therefore, (101110010)2 is now (0001)2 (0111)2 (0010)2. Now, we have the following: Decimal equivalent of (0001)2 = 1 Decimal equivalent of (0111)2 = 7 Decimal equivalent of (0010)2 = 2 (101110010)2 = (172)16
10^n Method (squaring)
B+2x | x^2
cubing
B+3x || 3.x² || x³ 104 100+12 || 48 || 64 11,24,864 unit digit of cube from *0-9*
Divisibility Rules
For *3* If sum total of all the digits is divisible by 3, then the number will be *÷ by 3* For *4* If the *last two digit* of a number is divisible by 4, For *5* If the last digit of the number is *5 or 0*,. For *6* If the last digit of the number is *divisible by two* and sum total of all the digits of number is *divisible by 3*, For *7* 371: 37 − (2×1) = 37 − 2 = 35; 3 − (2 × 5) = 3 − 10 = −7; thus, since −7 is divisible by 7, 371 is divisible by 7. For *8* If the *last 3 digits* of number is divisible by 8, For *9* If the *sum of digits* of the number is divisible by 9, For *11* A number is divisible by 11, if the *difference between* the sum of the digits at even places and the sum of the digits at odd places is divisible by 11 For *12* If the number is *÷ by 3 and 4*,
Number of Dividers
If N is any number that can be factorized like *N = a^p × b^q × c^r ×...,* where a, b, and c are prime numbers, then the number of divisors = *(p + 1) (q + 1) (r + 1)*
Basics of Remainder
If any positive number A is divided by any other positive number B and if *B > A, then the remainder will be A* itself. In other words, if the numerator is smaller than the denominator, then the numerator is the remainder
Fermat's Remainder Theorem
Let P be a *prime number* and N be a number non-divisible by P. Then, remainder obtained when A^(P−1) is divided by P is 1. (The remainder obtained when *A^(P−1)/P = 1*, if HCF (A, P) = 1.) What is the remainder when 2¹⁰⁰ is divided by 101? Solution Since it satisfies the Fermat's theorem format, remainder = 1. 2. (A+1)ⁿ/A when N is even, remainder is 1 3. Aⁿ/(A+1) N- even R=1; N= odd R=A 4. *(aⁿ + bⁿ) is divisible by (a + b)*, if *n is odd*. The extension of the abovementioned formula (aⁿ + bⁿ + cⁿ) is divisible by (a + b + c), if n is odd and a, b, and c are in arithmetic progression 5. (aⁿ -bⁿ) is divisible by (a + b), if *n is even*. (aⁿ − bⁿ) is divisible by (a − b), if n is even or odd
Half yearly compounding:
R/2, 2×n
Expression for Simple Interest and Compound Interest
SI=Pnr/100 CI= p(1+r/100)ⁿ
Extension of Product Stability Ratio
So, 1/50 ↑ corresponds to 1/51 ↓. It means that if we increase A by 2%, then B is needed to be decreased by 1.96% (approx.) so that P remains constant.
PRODUCT STABILITY RATIO
Speed × Time = Distance Price × Consumption = Expenditure *Number of persons × Days = Work done* Length × Breadth = Area of rectangle *A × B = P* Now, if A is increased by a certain percentage, then B is required to be decreased by certain percentage so that the product *(P) remains stable*. For example, if we increase A by 25% and P has to be constant, then B is required to be decreased by 20%
Process to Find HCF
Step 1 Factorize all the numbers into their prime factors. Step 2 Collect all the common factors. Step 3 Raise each factor to its minimum available power and multiply. Example 7 HCF of 100, 200, and 250 Solution Step 1 100 = 2² × 5²; 200 = 2² × 5²; 250 = 5³ × 2¹ Step 2: 2, 5 Step 3 *2¹ × 5²* = 50
Process to Find LCM
Step 1 Factorize all the numbers into their prime factors. Step 2 Collect all the distinct factors. Step 3 Raise each factor to its *maximum available power and multiply*. Example LCM of 10, 20, 25. Solution Step 1 10 = 2¹ × 5¹; 20 = 2² × 5₁; 25 = 5² Step 2 2, 5 Step 3 *2² × 5²* = 100 we have to know which factor of 25 is not present in 20; then, we need to multiply it by this factor. Therefore, 25 is having 5² and 20 is having 5¹ only, and hence, we will *multiply 20 by 5.*
Successive % change
Suppose we have to increase a quantity successively by 20% and 30%. . It can be seen below: 100 → 20%↑ → 120 → 30%↑→ 156 So, net percentage increase = 56% This is known as straight line method of solving the problems. Alternatively, * (a+b+ab/100)%*
Highest Common Factor (HCF)
The factors that are positive integral values of a number and can divide that number is called HCF. HCF, which is also known as *Greatest Common Divisor (GCD)*, is the highest value that can divide the given numbers. Therefore, 10 will be the HCF of 20 and 30.
Number Line
The number line is used to represent the set of real numbers. The following is the brief representation of the number line:
Remainder Theorem Method
The product of any two or more than two natural numbers has the same remainder when divided by any natural number as the product of their remainders. Remainder 12×13/7 × = Remainder 156/7 Remainder 12/7 = 5 Remainder 13/7 = 6 Remainder 12×13/7 = *(5 × 6)/7* = Remainder 30/7 = 2
Units Digit Method (multiplication)
This method of multiplication uses the sum of the unit's digit, provided all the other digits on the left-hand side of the unit digit are the same. 62*63= (6+0.5)x6 | (2+3) 39 | 06 3906
determination of prime factors and composite factors
number of composite factors, we will subtract the number of prime factors and 1 from the total number of factors. Example 22 Find the number of prime factors and composite factors of N = 420. Solution 420 = 2² × 3¹ × 5¹ × 7¹ Number of prime factors = 4 (namely 2, 3, 5, 7). Total number of factors = (2 + 1) (1 + 1) (1 + 1) (1 + 1) = 3 × 2 × 2 × 2 = 24 Therefore, the total number of composite factors = total number of factors − prime factors − 1 = 24 − 4 − 1 = 19.
Avg of squares of n consecutive even no.
[2(n+1)(2n+1)]/3
average and sum of a. 1st n consecutive even natural numbers b. 1st n consecutive odd natural numbers
a. n+1, n(n+1) b. n, n²
Cyclicity method (remainder)
for every expression of the remainder, there comes attached a specific cyclicity of remainders. Example 26 What is the remainder when 4¹⁰⁰⁰ is divided by 7? Solution To find the cyclicity, we keep finding the remainders until some remainder repeats itself. It can be understood with the following example: Number/7—4¹ 4² 4³ 4⁴ 4⁵ 4⁶ 4⁷ 4⁸ Remainder—4 2 1 4 2 1 4 2 Now, 4⁴ gives us the same remainder as 4¹ therefore, the cyclicity is of 3 (this is because remainders start repeating themselves after 43). Thus, any power of 3 or a multiple of 3 will give a remainder of 1, and hence, 4⁹⁹⁹ will give 1 as the remainder. Final remainder = 4
Time-Work: INDIVIDUAL WORK AND INDIVIDUAL EFFICIENCY
graphical method LCM method
perfect number
is a positive integer that is equal to the sum of its proper positive divisors, that is, the *sum of its positive divisors excluding the number itself* Equivalently, a perfect number is a number that is *half the sum of all of its positive divisors* (including itself) i.e. σ1(n) = 2n.
least common multiple (LCM)
of two numbers is the *smallest positive number that is a multiple of both*. Let us explain it through an example: LCM of 10, 20, and 25 is 100. It means that 100 is the lowest number, which is *divisible by all these three numbers*. LCM is a concept defined only for *positive numbers*, whether the number is an integer or a fraction. In other words, LCM is *not defined for negative numbers or zero*.
Number theory
or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the *study of the integers*. It is sometimes called *"The Queen of Mathematics"* because of its foundational place in the discipline. Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers).
Argand diagram
the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the *imaginary part by a displacement along the y-axis*.
Multiplication
the multiplier is written first and multiplicand second, "3 times 4" 3×4
What percentage of x is y?
⇒ y × 100÷x