Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

MEDIUM

Earn 100

Set of residue classes modulo $5$ is

50% studentsanswered this correctly

Important Questions on Modular Arithmetic

HARD

Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

Let

a_{1} = 24

and form the sequence

a_{n}, n \geq 2

by

a_{n} = 100 a_{n - 1} + 134

. The first few terms are

24, 2534, 253534, 25353534, \dots . . . .

. What is the least value of

n

for which

a_{n}

is divisible by

99

?

EASY

Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

Let

*

be a binary operation on the set

R

, of all real numbers defined by

5 (a^{*} b) = a b

, for all

a, b \in R

. The inverse of

0.5

under the operation

*

is

HARD

Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

For

n \geq 1,

let

a_{n}

be the number beginning with

n 9's

followed by

744;

e.g.,

a_{4} = 9999744 .

Define

f (n) = \max \{m \in N ∣ 2^{m} divides a_{n}\}

for

n \geq 1 .

Find

f (1) + f (2) + f (3) + \dots \dots . + f (10)

EASY

Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

Examine whether the operation

*

defined on

R

by a

a * b = a b + 1

is a binary or not.

EASY

Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

If

A = \{a, b, c\}

, then the number of binary operations on

A

is

EASY

Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

If

a * b

denotes the larger of

' a'

and

' b'

and if

a \circ b = (a * b) + 3

, then write the value of

(5) \circ (10)

, where

* and \circ

are binary operations.

HARD

Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

Let

T

be the smallest positive integer which, when divided by

11, 13, 15

leaves remainders in the sets

{7, 8, 9}, {1, 2, 3}, {4, 5, 6}

, respectively. What is the sum of the squares of the digits of

T ?

EASY

Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

Which of the following is not a group?

EASY

Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

If

a \equiv b (m o d m)

and

x

is an integer, then which of the following is incorrect?

EASY

Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

Let* be a binary operation on the set

R

of real numbers defined by

a^{*} b = \frac{3 a b}{7}

, then the identity element in

R

for

^{*}

is

EASY

Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

The binary operation

* : R \times R \to R

is defined as

a * b = 2 a + b

. Find

(2 * 3) * 4

MEDIUM

Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

For the binary operation

*

defined on

R - \{1\}

by the rule

a * b = a + b + a b

for all

a, b \in R - \{1\}

, the inverse of

a

is

EASY

Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

Determine whether the binary operation $*$ on $R$ defined by $a * b = | a - b |$ is commutative. Also, find the value of $(- 3) * 2$ .

HARD

Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

A five-digit number

n = \bar{a b c d e}

is such that when divided respectively by

2, 3, 4, 5, 6

the remainders are

a, b, c, d, e .

What is the remainder when

n

is divided by

100 ?

EASY

Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

In the set of integers under the operation $*$ defined by $a * b = a + b - 1$ , the identity element is

EASY

Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

Prove that

{(a^{- 1})}^{- 1} = a

for every

a \in G

, group.

EASY

Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

Prove that the identity element of a group is unique.

EASY

Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

The number of binary operations on the set

{1, 2, 3}

is

MEDIUM

Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

If

*

is a binary operation defined by

a * b = \frac{a}{b} + \frac{b}{a} + \frac{1}{a b}

for positive integers

a

and

b

, then

2 * 5

is equal to

EASY

Mathematics>Arithmetic>Modular Arithmetic>Basics of Modular Arithmetic

On the set of positive rationals, a binary operation

*

is defined by

a * b = \frac{2 a b}{5} .

If

2 * x = 3^{- 1}

then

x =