Previous Year Questions on Integers

Integers - Previous Year Questions

Previous Year Questions on Integers

Previous year questions on Integers cover one of the oldest and most fundamental topics in mathematics. Integers, a subset of real numbers, have been studied since ancient times for their applications in arithmetic and algebra. They are crucial in understanding divisibility, prime numbers, and modular arithmetic. Explore more in Mathematics Notes.

Vidyasagar University

2023-24 (CBCS)

Use Euclid algorithm to find inetgers \(u\) and \(v\) satisfying \(72u+42v=132\). [2]
Find the remainder when \(1!+2!+3!+…+1000!\) is divided by \(24\). [2]
For \(a,b\in \mathbb{Z}\), \(m\in \mathbb{Z}^{+}\), if \(a\equiv b(mod~ m) \), then show that \(f(a)\equiv f(b)(mod~ m) \), where \(f(x)\) be a polynomial of degree \(n\) with integer coefficients. [2]
Find the unit digit in \(77^{77} \). [2]
Show that the product of any \(p\) consecutive integers is divisible by \(p\). [5]
Use Division algorithm to show that the square of an odd integer is of the form \(8k+1,~k\in \mathbb{Z}\). [4]
If \(a,b\in \mathbb{Z}\), not both zero, and \(k\in \mathbb{Z}^{+}\), then show that \(gcd(ka,kb)=k.gcd(a,b) \). [3]
Prove that for any two integers \(U\) and \(V\), there exists two unique integers \(m\) and \(n\) such that \(U=mV+n\), \(0\leq n \lt V\). [4]
If \(a\equiv b(mod~ m) \) and \(a\equiv c(mod~ n) \), prove that \(b\equiv c(mod~ d) \) where \(d=gcd(m,n) \). [4]
Prove that \(n^{2}+2\) is not divisible by \(4\) for any integer \(n\). [2]
Find the number of divisors and their sum of \( 10800\). [2]

2021-22 (CBCS)

Prove that \( 6|n(n+1)(n+2), ~n\in \mathbb{Z}\). [5]
Use the theory of congruence for finding the remainder when the sum \(1^{5}+2^{5}+3^{5}+…+100^{5}\) is divided by \(5\). [5]
If \(s_{n}=1+\frac{1}{2}+\frac{1}{3}+…+\frac{1}{n} \) then prove that \(s_{n}\gt \frac{2n}{n+1}\) if \(n\gt 1 \). [3]
Show that \((2n+1)^{2}\equiv 1(mod~ 8)\) for any naturan number \(n\). [3]

2020-21 (CBCS)

If \(a\) is prime to \(b\), prove that \(a+b\) is prime to \(ab\). [2]
If \(n\) be a positive integer and \((7+2i)^{n}=a+ib \), then prove that \(a^{2}+b^{2}=53^{n} \). Hence express \(53^{2} \) as the sum of two squares. [2]
If \(2^{n}-1\) be a prime, prove that \(n\) is a prime. [2]
If \(n\) be a positive integer greater than \(2\), then prove that \( (n!)^{2}\gt n^{n}\). [2]
Prove that the product of any \(m\) consecutive integers is divisible by \(m\). [5]
Prove that for any two integers \(a\) and \(b\), \(a\equiv b(mod~ m) \) if and only if \(a\) and \(b\) leave the same remainder when divided by \(m\). [6]

2019-20 (CBCS)

Use the \(2^{nd}\) principle of induction to prove that \(\left(3+\sqrt{7} \right)^{n}-\left(3+\sqrt{7} \right)^{n} \) is an even integer for all \(n\in \mathbb{N} \). [2]
Find the remainder when \(777^{777}\) is divided by \(16\). [2]
If \(k\) be a \(+ve\) integer, show that \(gcd(ka,kb)=k~gcd(a,b)\). [2]
Use division algorithm to prove that square of any integer is of the form \(5k\) and \(5k+1\), \(k\in \mathbb{Z}\). [2]
Use theory of congruences to prove that \(17|2^{3n+1}+3.5^{2n+1}\), \(\forall n\geq 1 \). [2]
Prove that the product of any \(m\) consecutive integers is divisible by \(m\). [4]
State the Fundamental theorem of Arithmetic. [1]

2018-19 (CBCS)

Find integers \(u\) and \(v\) satisfying \(52u-91v=78\). [2]
Using the principle of induction, prove that \(2.7^{n}+3.5^{n}-5\) is divisible by \(24\) for \(n\in \mathbb{N}\). [2]
Find the remainder when \(1!+2!+3!+…+50!\) is divided by \(15\). [2]
If \(a\) is prime to \(b\) and \(a\) is prime to \(c\) then prove that \(a\) is prime to \(bc\). [2]
Find the unit digit in \(7^{99} \). [2]
Use division algorithm to prove that square of an odd integer is of the form \(8k+1\) where \(k\) is an integer. [3]
Use Euclid algorithm to find inetgers \(u\) and \(v\) satisfying \(gcd(72,120)=72u+120v\). [2]

2017-18 (CBCS)

Prove that square of any integer is of the form \(3k\) or \(3k+1\). [2]
Prove that, there exists no integer in between 0 and 1. [2]
If \(s_{n}=1+\frac{1}{2}+\frac{1}{3}+…+\frac{1}{n} \) then prove that \(s_{n}\gt \frac{2n}{n+1}\) if \(n\gt 1 \). [2]
State the Fundamental theorem of Arithmetic. [1]
If \(a\) divides \(b\), then prove that every divisor of \(a\) divides \(b\). [1]
Prove that \(1^{n}-3^{n}-6^{n}+8^{n}\) is divisible by \(10\) \(\forall n\in \mathbb{N}\). [2]
Find integers \(u\) and \(v\) satisfying \(20u+63v=1\). [3]
State the division algorithm on the set of integers. [1]
Find integers \(s\) and \(t\) such that \(gcd(341,1643)=341s+1643t\). [2]
Use the theory of congruence for finding the remainder when the sum \(1^{5}+2^{5}+3^{5}+…+100^{5}\) is divided by \(5\). [2]

FAQs

What are integers in mathematics?
Integers include all whole numbers, both positive and negative, along with zero.
What is the importance of integers?
Integers are fundamental in mathematics and are used in counting, ordering, and various algebraic operations.
What are prime numbers?
Prime numbers are integers greater than 1 that have only two divisors: 1 and themselves. Learn more in Mathematics Notes.
What is modular arithmetic?
Modular arithmetic involves calculations with remainders and is widely used in computer science and cryptography.
What is the GCD of two integers?
The greatest common divisor (GCD) is the largest integer that divides both numbers without leaving a remainder.
How is LCM calculated?
The least common multiple (LCM) is the smallest integer that is a multiple of two or more integers.
What are composite numbers?
Composite numbers are integers greater than 1 that have more than two divisors.
How are integers used in everyday life?
Integers are used in counting, measuring, financial calculations, and digital encoding.
What is the difference between integers and natural numbers?
Natural numbers are positive integers starting from 1, while integers include negative numbers and zero.
Why are integers important in number theory?
Integers are studied in number theory to understand divisibility, primes, and modular systems. Explore more in Mathematics Notes.