How many positive integers N satisfy all of the following three conditions? (i) N is divisible by 2020. (ii) N has at most 2020 decimal digits. (iii) The decimal digits of N are a string of consecutive ones followed by a string of consecutive zeros.

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A: We know,  abundant number is a number that is smaller than the sum of its proper divisors. An…

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A:

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: Find a positive integer n for which the equality holds = 1 2

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Q: Prove that for every non-negative integer n the value of n² + n+41 is prime ?

A: The number n² + n + 41 is not prime for every natural number n. Let, f(n) = n² + n + 41

Q: . Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when…

A: Consider an integer x that has a remainder of 2 when it is divided by 5 Thus x=2 mod 5 0r x=5k+2 It…

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Q: Prove by contradiction : For any odd integer n : n² – 7 is not divisible by 4.

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Q: 6. Use contradiction to prove that, for all integers k > 1, 2/k+1+ 2/K+2

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Q: Find all integers n > 1 such that 1n − 1)" is +2+...+(n divisible by n.

A: To find all the integers n>1 such that 1n+2n+...+n-1n is divisible by n

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Q: 1. Prove that the negative of any even integer is even. 2. Prove that the difference between an even…

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Q: Prove that the last two digits of 2" cannot be 02 and the last three digits cannot be 108. Find all…

A: All these problems can be solved using modular arithmetic. We presented the detailed solution in the…

Q: The sum of any two integers of the from 4k + 1is always: Even Odd prime Zero Windows b

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Q: Find all integers n such that n| (2n+3). *

A: Given, n|(2n+3)     TO find, All integers n

Q: Use cases to prove that for all integer n, n2+3n+5 is odd.

A:

Q: 4. Find all the prime numbers that can be expressed in the form n2 – 9 for some integer n.

A:

Q: n any positive integer n, there exist n consecutive composite integers

A: Given that given any positive integer n, show that there exist n consecutive composite integers.

Q: 21. How many positive integers x are there such that 3x has 3 digits and 4x has four digits?

A:

Q: snip

A: To Prove n! ≥ 2n for all intergers n≥4We may prove this by Induction Method.

Q: Prove that 6 divides n3-n for all non-negative integers n.

A: See the detailed solution below.

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

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Q: Determine all integers n ≥ 3 such that Cn is planar.

A: ANSWER :

Q: Find all positive integers n such that 2" – 1 | 3" – 1. - -

A: Given 2n-1|3n-1 ,  for n∈ℤ+ To find if the statement is true.

Q: Let n be a positive integer. Prove that :)- (:)- +2. +...+n. = n2"-1

A: Binomial Theorem: According to the binomial theorem, let x and y are any two numbers. Then it is…

Q: 2. What are the two largest positive integers n satisfying n - 1 | n²-A?

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Q: prove that n²_1 is divisible by when ever n is odd an Positive integer.

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

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

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Q: By using mathematical induction prove that the given equation is true for all positive integers. 9n…

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Q: Find all integers n> 1 such that 1" + 2"+... +(n – 1)" is divisible by n.

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Q: Find all positive integers n such that the sum of all positive divisors of n is odd.

A: Given the data

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Q: Prove that for any odd positive integer, n, the sum of n consecutive positive integers is divisible…

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

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Q: Find all positive integers n such that p(n)|n.

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Q: 3.3. Proof that for all integers numbers n, if n3 is odd then n is odd, using proof by…

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Q: Find all positive integers n such that both n + 2020 and n + 2021 integers. are LVn +1]

A:

Q: 3. Prove that if n is odd, then 8 divides n2- 1. Prove that 6 divides n п. 4. Let n and k be two…

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Q: 24. Suppose that I choose six of the first ten positive integers. Prove that I must have chosen two…

A: Rewrite the first ten numbers as follows.

Q: 74. P is a natural number of at least 6 digits and its left- most digit is 7. When this leftmost…

A: Introductions :

Q: Prove by direct method: for any n integer,if n is odd then 3n+2 is odd

A:

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

A: Given: 1+3+5+...........+2n-1 is n2

Q: Prove by induction that the product of the first n even numbers is equal to n!2" for all positive…

A: We have to prove by induction that the product of first n even numbersis equal to n!2n for all…

Q: Make a table of values of the numbers 10n – 1 and 10n + 1 for 1 < n < 20. Circle any numbers that…

A: Table of values for 10n-1 and 10n+1 for n between 1 and 20. n 10n-1 10n+1 1 9 11…

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

A:

Q: 5. Find all integers 100 ≤ n ≤ 999 such that the last three digits of n² is same as n.

A: To find: All integers  100≤n≤999 such that the last three digits of n2 is same as n.

Percentage

A percentage is a number indicated as a fraction of 100. It is a dimensionless number often expressed using the symbol %.

Algebraic Expressions

In mathematics, an algebraic expression consists of constant(s), variable(s), and mathematical operators. It is made up of terms.

Numbers

Numbers are some measures used for counting. They can be compared one with another to know its position in the number line and determine which one is greater or lesser than the other.

Subtraction

Before we begin to understand the subtraction of algebraic expressions, we need to list out a few things that form the basis of algebra.

Addition

Before we begin to understand the addition of algebraic expressions, we need to list out a few things that form the basis of algebra.