MEDIUM
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Set of residue classes modulo 5 is

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Important Questions on Modular Arithmetic

HARD
Let a1=24 and form the sequence an, n2 by an=100an-1+134. The first few terms are 24, 2534, 253534, 25353534, ..... What is the least value of n for which an is divisible by 99?
EASY
Let * be a binary operation on the set R, of all real numbers defined by 5a*b=ab, for all a,bR. The inverse of 0.5 under the operation * is
HARD
For n1, let an be the number beginning with n9's followed by 744; e.g., a4=9999744. Define fn=maxmN2m divides an for n1. Find f(1)+f(2)+f(3)+.+f(10)
EASY
Examine whether the operation * defined on R by a a* b=ab+1 is  a binary or not.
EASY
If A=a,b,c, then the number of binary operations on A is
EASY
If a * b denotes the larger of 'a' and 'b' and if ab=(a*b)+3, then write the value of 510, where * and  are binary operations.
HARD
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

Which of the following is not a group?

EASY
If a≡b mod m and x is an integer, then which of the following is incorrect?
EASY
Let* be a binary operation on the set R of real numbers defined by a*b=3ab7, then the identity element in R for * is
EASY
The binary operation *: R×RR is defined as a*b=2a+b. Find 2*3*4
MEDIUM
For the binary operation * defined on R1 by the rule a*b=a+b+ab for all a, bR1, the inverse of a is
EASY

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

HARD
A five-digit number n=abcde¯ 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

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

EASY
Prove that a-1-1=a for every aG, group.
EASY
Prove that the identity element of a group is unique.
EASY
The number of binary operations on the set {1, 2, 3} is
MEDIUM
If * is a binary operation defined by a*b=ab+ba+1ab for positive integers a and b, then 2*5 is equal to
EASY
On the set of positive rationals, a binary operation * is defined by a*b=2ab5. If 2*x=3-1 then x=