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WorksheetsMIDTERM EXAMINDATION FOR NUMBER THEORY 2024
Total questions: 71
Worksheet time: 51mins
Branch of pure mathematics devoted primarily to the study of the integers.
Number Theory
Arithmetic
Cryptography
Aryabata
It is based on the idea that multiplying primes together is much easier than figuring out which primes were multiplied to form a number.
Arithmetic
Number Theory
Cryptography
Aryabhata
Positive whole numbers excluding zero, it is also called natural numbers.
Whole Numbers
Rational Numbers
Counting Numbers
Real Numbers
A number that can be written as fraction and whose numerator and denominators are integers.
Whole Numbers
Rational Numbers
Counting Numbers
Real Numbers
This formula is often used in proofs and derivations, which makes repeated use of the familiar factorial function.
Factorial Formula
Recursive Formula
Pascal’s Triangle
Roman Coefficient
Binomial coefficient of 7choose 5 using Recursive Formula.
21
22
23
24
A number is completely divisible by ______ if the sum of its digits is divisible by 3.
12
6
9
3
If subtracting twice of the last digit from the number formed by remaining digits is 0 or divisible by 7, the number is divisible by ______.
14
7
21
24
If a number is divisible by both 3 and 4, then the number is divisible by ______.
7
12
14
24
Identify a number divisible by 9.
117
998
921
89
Subtract the last digit from the number formed by the remaining digits is divisible by____?
3
5
7
11
Divisibility Theorem a=94, b=564. Find the value of c?
112
128
1168
1128
The pigeonhole principle is also termed as ______.
Dirichlet Principle
Drichilet Principle
Gustav Lejeune
Mathematical Induction
24 students enrolled in BSMT 2-1 this year, at least how many students are born in the same month?
8
6
4
2
If n items are put into m containers, with n>m, then at least one container must contain more than one item.
Pigeonhole Principle
Drichilet Principle
Mathematical Induction
Lejeune Principle
How many cards must be selected from a standard deck of 52 cards to guarantee that at least three cards of the same suit are chosen?
3
6
9
12
Suppose there are 35 different time periods during which classes at the local college can be scheduled. If there are 210 different classes, what is the minimum number of rooms that will be need?
6
12
24
36
Concept that helps to prove mathematical results and theorems for all natural numbers.
Dirichlet Principle
Drichilet Principle
Gustav Lejeune
Mathematical Induction
Steps to solve mathematical induction are follows, except
Base step
Proving Step
Assumption Step
Induction Step
Assume that P (k) is true for some n=k.
Induction Step
Assumption Step
Proving Step
Base Step
Which step in mathematical induction is a statement of fact?
Assumption Step
Base Step
Proving Step
Induction Step
Which step in mathematical induction is a conditional one and also known as Induction Hypothesis?
Base Step
Assumption Step
Induction Step
Proving Step
Mathematical Induction step 2. Given : 1+3+5+…+(2n-1)=n²
1+3+5+…+(2k-1)=k²
1+3+5+…+(2k+1)=k²
1+3+5+…+(2k-1)=2k
1+2+3+5+…+(2k-1)=k²
Mathematical Induction step 3. Given : 1+3+5+…+(2n-1)=n²
(k+1)²=(k+1)²
(k+1)=(k+1)²
(k-1)²=(k-1)²
(k-1)=(k-1)²
Mathematical Induction step 1. Given : 1+2+2²+…+2⁴=2⁵-1
P (1) is True
P (1) is False
P (1) is Infinite
P (1) is Undefined
Sum of all composite numbers between 25 and 30.
78
62
87
It is a positive integer having exactly two factors, 1 and itself.
Prime Number
Composite Number
Rational Number
Counting Number
He discovered prime numbers.
Plato
Eratosthenes
Archimedes
Leonard Euler
The common method used prime factorization method.
Factor Tree Method
Family Tree Method
Exponential Method
Multiplication Method
Find the HCF of 50 and 70 using factor tree method
5
7
10
2
Find the congruence class of 7 modulo 3.
2
4
6
1
Find the congruence class of 12 modulo 5.
7
3
5
2
It is generally refers to the mathematical concept where two objects or shapes are the same in shape and size.
Fundamental Theorem of Arithmetic
Theory of Congruence
Euclidean Algorithm Primes
Sieve of Eratosthenes
It refers to the number by which an integer is divided to determine its remainder.
Modulus
Modulo
Integers
Prime Numbers
It is a way to categorize integers based on their remainders when divided by a specified modulus.
Modulus
Modulo
Prime Numbers
Integers
In a given a 7 mod 3 = 1, "1" is the modulus.
True
False
Uncertain
Undefined
In a given a 7 mod 3 = 1, "3" is the modulus.
True
False
Uncertain
Undefined
Every element is equivalent to itself.
Reflexivity
Symmetry
Transitivity
Identity elements
If a ≡ b (mod m), then b ≡ a (mod m) because they leave the same remainder
Reflexivity
Symmetry
Transitivity
Identity elements
: (8 Ξ 29 mod 7) and
: (29 Ξ 15 mod 7 then
: (8 Ξ 15 mode 7)
Addition Property
Transitive Property
Symmetric Property
Reflexive Property
: (10 Ξ 1 mod 9) then
: (102 Ξ 12 mod 9)
: (100 Ξ 1 mod 9)
Addition Property
Multiplication Property
Exponential Property
Symmetric Property
: If (8 Ξ 3 mod 5) and
: (40 Ξ 15 mod 5 then
: (8 x 40 Ξ 3 x 15 mod 5)
: (320 Ξ 45 mod 5)
Addition Property
Multiplication Property
Exponential Property
Transitive Property
The following are the basic properties of congruence, except
Addition Property
Multiplication Property
Division Property
Exponential Property
The following are the parts of theory of congruence, except
Integer
modulus
modulo
prime
In a given a 109 mod 7 = 4, "4" is the integer
True
False
Uncertain
Undefined
Eratosthenes sieve filters out composite numbers to find the prime numbers.
True
False
Uncertain
Undefined
It is the smallest number that is a multiple of two or more given numbers.
LCM
GCF
HCF
Prime Factor
Least Common Multiple of numbers can be calculated using various methods, except
Listing Method
Prime Factorization
Division Method
Factor Tree
The following are the steps in finding the LCM using listing method, except
List the first few multiples of A and B
Mark the common multiples from the multiples of both numbers
Select the Smallest common multiple, that lowest common multiple is the LCM of the numbers
Select the Largest common multiple, that lowest common multiple is the LCM of the numbers
Steps in finding the LCM using Prime Factorization Method, except
Find the quotient of these numbers with the highest powers is the LCM of the given numbers.
Find the prime factors of the given numbers by repeated division method.
Write the numbers in their exponent form. Find the product of only those prime factors that have the highest power
The product of these factors with the highest powers is the LCM of the given numbers.
Relatively prime integers, also known as coprime or mutually prime integers, are numbers that share a common divisor.
True
False
true more than 1 common divisor
False, no common divisor other than itself
Smallest Prime number
0
1
2
3
Largest Prime Number
2(99,589,933)-1
2(82,589,933)-1
2(82,589,933)-2
2(99,589,933)-2
Twist Prime Numbers, except
13 and 31
17 and 71
79 and 97
19 and 91
Sum of prime numbers less than 50
323
328
217
289
Mathematical Induction: Find the Assumption
1 + 2 + 3 + 4 + ... + n = 2n(n+1)
1 + 2 + 3 + 4 + ... + n = 2k(k+1)
1 + 2 + 3 + 4 + ... + n = 2k(k−1)
1 + 2 + 3 + 4 + ... + n = 2(k+1)
1 + 2 + 3 + 4 + ... + k = 2k(k+1)
The following are methods for finding GCD, except
Listing Out the Factor
Long Division Method
Euclid's Algorithm
Sieve of Eratosthenes
It is use for finding the greatest common divisor between two numbers. It's uses the principle that the GCD of a set of two numbers does not change if you replace the larger of the two with the remainder when you divide the larger of the two by the smaller.
Euclid's Algorithm
Eratosthenes Algorithm
Euclidean Theory
Eratosthenes Theory
Euclid's Algorithm
Multiplicand= Multiplier x Product - Remainder
Dividend= Divisor x Quotient + Remainder
Divisor = Dividend x Quotient + Remainder
Multiplier = Multiplicand x Product + remainder
Find the GCD of 126, 162, and 180
16
18
8
6
Basic Properties: Residue Classes are the following, except
Associative Properties
Transformative Properties
Commutative Properties
Distributive Properties
