Let a be a positive integer. Then lcm(2a+1,20+2)= *

Q: Let i be a positive integer and A, [-i+ 1,1]. Then U14, %3D O R O None of these O F1,11

A: Given that 'i ' be a positive integer and Ai=-i+1,1. We have to find the union ∪i=1∞ Ai.

Q: Find an expression for An, where n is an arbitrary positive integer.

A: Given: A=1423∈M2×2R

Q: b.) Show that if a and b are positive integers where a is even and b is odd, then (a,b) = (a/2, b).…

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Q: Let a be an odd integer. Show that (i) 8 | a2 − 1; (ii) 16 | a4 + 15.

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Q: Prove: For all integers a,b and c, if a|b and a|c, then a|(3b–2c)

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Q: Let a be a positive integer. Then lcm(2a+3,2a+4)= O (2a+3)(a+2) (2a+3)(2a+4) 9a None of these O…

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Q: a|b then albc for all integers с.

A: Statement is ture

Q: Let a be a positive integer. Then Lcm(2a+1,2a+2)= *

A: The given numbers are 2a+1 and 2a+2. Clearly, the given numbers are two consecutive numbers.

Q: For every positive integer n, prove that 7n – 3n is divisible by 4.

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Q: Let a be a positive integer. Then lcm(3a,3a+1) = 9a None of these O a(3a+1) О За(За+1) O a(a+1)

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Q: Let a and b be integers so that (a, b) = 1. Prove that (a + b, a – b) divides 2.

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Q: For every positive integer a, find a composite number n such that n|a^n - a.

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Q: Every positive integer is the sum of at most two squares and a cube of nonnegative integers. O True…

A: consider 0  0=02+02+03 now  1 can be written as  1=02+02+13 now consider 2 2=12+02+13 now consider 3…

Q: Let a be an algebraic integer in Q(v-37) and let A = (2, 1 + v-37). Prove that either a or a –- 1 is…

A: Given that,         α is an integer in Q-37 lets consider  α=-37⇒α-1=-37-1

Q: 5. Let a be a real number, c an integer, c > 0. Prove that

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Q: Let a be a positive integer. Then lcm(2a-1,2a)= O None of these O a(2a-1) O a(a+1) O 4a O 2a(2a-1)

A: a is a positive integer.

Q: If a, , and c are integers such that albc, then alb or alc. :Select one True False

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Q: if a is a positive integer less than 10, then 7a7,7a7,7a7 is divisible by 3 Select one: O True O…

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Q: For an arbitrary integer a, verify the following: (a) 2|a(a + 1), and 3| a(a + 1)(a + 2).

A: We have to verify, 2|a(a+1)  and 3|a(a+1)(a+2) Here we use division algorithm.

Q: Let i be a positive integer and A, = [-i+1,1]. Then U2, A

A: We have to use properties of union in this question.

Q: 4. Let a be an odd integer. Show that (i) 8| a² – 9; (ii) 16 | aª – 81.

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Q: If all entries of A and A -1 are integers, prove that <let A = 1 or -1. Hint: What is det A times…

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Q: If z ≥ 2 is an integer, show that the following numbers are composite. n! + 2, n! + 3, n! + 4, ... ,…

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Q: If a is any integer, is a(a + 1) even or odd?

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Q: Let a be a positive real number and let n be an integer greater than 1.Prove or disprove that…

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Q: a.) Let a and b are positive integers. Prove that (a,b) = 2(a/2, b/2).

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Q: Let a, b be nonzero integers with (a, b) = 1. Compute (a + b, a – b).

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Q: For an arbitrary integer, say A, verify that 2|A(A+1). Similarly, verify that 3|A(A+1)(A+2).

A: To verify:A(A+1) is divisible by 2.

Q: If a, and c are integers such that albc, then alb or alc. :Select one True False

A: Given integers 'a' 'b' and 'c' such that a|bc then a|b or a|c. This statement is false.  Let us…

Q: Find all positive integers a, b such that a | b^2-b+1 and b | a^2-a+1

A: Solution :-

Q: If a is divisible by b, then bq, for some integer q. las

A: If a is divisible by b, then, a = bq for some integer q.  TRUE

Q: Prove or disapprove that if a|bc, where a,b,c are positive integers and a≠0, then a|b or a|c

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Q: prove that Every positive integer n can be written as n =( 2^a)(5^b)c, where c is not divisible by 2…

A: Fundamental Theorem of Arithmetic: Every positive integer (except n=1) can be uniquely expressed as…

Q: Let a be a positive integer. Then lcm(3a,3a+1)= O a(a+1) О а(За+1) O 9a O 3a(3a+1) ООО

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Q: 8. Let a, b, and c be integers. The show that alb a|bc

A: a, b & c are integers.

Q: Let p = p1 ... Ps be an integer with digits p1,..., Ps. Show that 0.p1 . . · Ps %3D 10° – 1

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Q: For all positive integers a, b, where a < b, if a {b, then gcd(a, b) = 1. O True False

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Q: Show that if a,b, and c≠0 are integers, then a∣b if and only if ac∣bc.

A: We have to prove the following statement -  If a,b, and c≠0 are integers, then a∣b if and only if…

Q: Let i be a positive integer and A = [-i + 1 ,1]. Then UA,

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Q: For every positive integer a, find a composite number n such that n|a" – a.

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Q: 11. Prove that for each integer a, if a? - 1 is even, then 4 divides a?

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Q: Let i be a positive integer and A, = [-i+1,1[. Then N14= %3D

A: We have to find the first intersection first.

Q: Let n be a positive integer and M %3D Then M"

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Q: Show that for any integer n ≥ 2, C(2n,n) > C(2n,n-1)

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Q: Let a be a positive integer. Then lcm(3a,3a+1)= * None of these За(За+1) a(За+1) 9a a(a+1)

A: Write all factor without repeating term

Q: Prove: For all integers a, b, and c, if alb and alc then a (2b+3c)

A: prove is given below ....

Q: Show that if n is an integer, thenn= [n/2] + [n/2].

A: Floor function x: x gives as output the greatest integer less than or equal to x. Ceiling function…

Q: rove that if x and y are odd positive integers ,then x+y is even but not divisible by 4

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Q: Show that if a | b and b | a, where a and b are integers, then a = b or a = −b.

A: We show that if a | b and b | a, where a and b are integers, then a = b or a = −b.

Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.