, Let a be a positive integer, where the positive difference of every two digits of a in base 10 are all written on a board. If some of the numbers are erased and only the numbers 2, 0, 2 and 2 are left, what is the least possible value of a? (A) 1011 (B) 1112 (C) 1113 (D) 2000 (E) 2022 D B

Q: 11. Prove that for each integer a, if a² – 1 is even, then 4 divides a² – 1.

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Q: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

A: We have to find how many 5’s does the prime factorization of x + y contain.

Q: If n is a positive integer with (n + 1)(n + 3) is odd, then (n + 2)(n+ 4) must be a multiple of 1) 6…

A: Given:  (n+1)(n+3) is odd We will get an odd number from product of two numbers only if the two…

Q: (4) Prove in two different ways that if a is a positive r number, then (a + 1)" > a" + na"-1:

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Q: Let yk = k", where n is a positive integer; then Ak" = (k + 1)" – k" - (1.112) nk"-+ n kn-2 +...+ 2.…

A: We have to solve given problem:

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

A: As per our guidelines we are supposed to answer only one question. Kindly repost other question as…

Q: 11. What is the sum of all possible positive integers a, such that for any positive is an integer?…

A: Claim: n(n+2)(n+4) is always divisible by 3. Proof: we know, n = 0 or 1 or 2 (mod 3)    When, n = 0…

Q: Prove that the sum of the first n cubes is equal to the square of the sum of the first n positive…

A: According to the given information, it is required to prove the following.

Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

Q: Let n ≥ 1 be an integer. Show that in any set of n consecutive integers, there is exactly one that…

A: According to the given information, it is required to show that in any set of n consecutive integers…

Q: 4. Prove that the difference between the square of any odd integer and the integer itself is always…

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Q: Suppose a set of 8 numbers are selected from the set: {1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13;…

A: It is given that 8 numbers are selected from the set 1,2,3,4,5,6,7,8,9,10,11,12,13,14.

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

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Q: Prove that 2^10 + 5^12 is a composite number.

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Q: In the 1989, Journal of Number Theory paper, N. Tzanakis and B. de Wegner showed that the largest of…

A: We have to factories the equation  x3+3x2+2x-258,474,216=0

Q: Let yk = k", where n is a positive integer; then Ak" = (k + 1)" – k" %3D (1.112) n = nk"-1 kn~2…

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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: 5) Let n be a positive integer such that 7 | (8-1). Prove that 7 | (8=*1 - 1). (Hint: Rewrite 8*1…

A: Given: Let n be a positive integer such that 78n-1. To prove: 78n+1-1

Q: Let, a1 = 3, a2 = 4 and for n > 3, an = 2an-1+an-2 + n², express an in terms of n.

A: In this question, we solve the recurrence relation  in terms of n an=2an-1+an-2+n2witha1=3a2=4

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

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Q: Let A be the set of all positive integers whose last two digits form a number that is evenly…

A: We have the given condition as   Let us consider A be the set of all positive integers whose last…

Q: n + 2020 n + 2021 and LVn +1]

A:

Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: Prove that the fourth power of any integer has one of the forms 5k or5k + 1, where k is an integer.

A: We need to prove that any integer has one of the forms 5k or 5k + 1, where k is an integer.Let p be…

Q: 8. If A= 0 1 then A ( where k is any integer number) is

A: 8. Given that A=1101 The objective is to find Ak where k is any integer number. Now,…

Q: 1 If n is a positive integer, then (D-a %3D n!

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Q: Show that if n is a positive integer, then (n + 1, n2 - n + 1) = 1 or 3

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Q: 9. (a) Show that if n is an integer, then: -o3* (") = 4". %3D (b) Let n be a positive integer. Show…

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Q: Show that for all integers a, b > 0, the number (a + b) – (a O b) is both nonnegative and even.

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Q: Show by contradiction that for any integer n: if n>0, then n+10 n+5

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Q: Consider all numbers between 20000 and 70000. Efficiently determine: A) How many of these numbers…

A: Part A - Let’s represent the number as ABCDE. You have to considere the special cases first, since…

Q: Let, a1 = 3, a2 = 4 and for n > 3, an = 2an-1+an-2 +n², express an in terms of n.

A: a1=3, a2=4 n≥3; an=2an-1+an-2+n2 First from the given, the given values shows that the terms are in…

Q: Let n = integer; 7 divides n if and only if 7 dividesn^2 (The result(s) ought to be integer(s))

A: Consider an integer n. We have to prove that 7 divides n if and only if 7 divides n2. Suppose that 7…

Q: show that no integer of the form n^3+1 is a prime other than 2

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Q: 2. Show that if a is an integer other than 0, then a) 1 divides a h) a divides 0

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

A:

Q: Let m and n be positive integers.

A: Given, m and n be positive integers.  To prove that mℤ⊕nℤ is an ideal of ℤ⊕ℤ and that…

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: If n is an even integer, then 3n° is an even integer.

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Q: prove that (n(nt)* s an integer V impers

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Q: Show that the sum of the first n positive odd integers,1 + 3 + 5 + g+ (2n - 1),is n2.

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Q: 9. Show that if n is a positive integer then n Σ k² ( x ) = n(n + 1)2²−² k k=0

A: Concept: Mathematical induction: Mathematical induction is a technique for proving that a statement,…

Q: .

A: The matrix                 A=12-12-12-12-1212-12-12-12-1212-12-12-12-1212 We have to compute A2 and…

Q: Compute (1+ v3)18 in F17[V3]. Your answer will lie in F17; input it as an integer between O and 16.

A:

Q: Prove that if a is a positive integer, then 4 does not divide a2 + 5.

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Q: a) Find the number of positive integers not exceeding 120 that are divisible by 3 or 4. 1 0 01 0 0 0…

A: As per our guides lines I answered only 1st part