Consider all positive integers with 3 different digits. (Note that 0 cannot be the first digit.) (1) How many are greater than 700? (ii) How many are dd? (iii) How many are even? (iv) How many are divisible by 5?

Q: Use a direct proof to show that the sum of two odd integers is even.

A: Types of integers : Odd integers : The integer which is not divided by 2 . It has the form 2k+1…

Q: Show that there are infinitely many primes of the form 4k + 3.

A: We have to show that there are  infinitely many primes of the form 4k+3.

Q: The number of factors of prime numberss are (None of the mentioned Depends on the prime number (2 (3

A: Prime number is only divisible by 1 and that number itself

Q: Consider all positive integers that are mueiples of 20 and that are less than or equal to 300. What…

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Q: 8. How many positive integers less than 1000 a) have exactly three decimal digits? b) have an odd…

A: According to our guideliness we can answer only the first 3 subparts. Positive integers less than…

Q: of the first 150 positive integers?

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

A: "Since you have posted a question with multiple sub-parts, we will solve the first three sub-parts…

Q: Which positive integers less than 20 are relatively prime to 20?

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Q: Suppose n ≥ 2 is an integer with the property that whenever a prime p divides n, p 2 also divides n…

A: let n=p1a1p2a2....pmam be the prime factorization of n  Here p1,p2....pm are all distinct primes and…

Q: How many integers greater than 5400 and less than 10000 have both of the following properties: -…

A: To count the number of positive integers satisfying all the required properties

Q: The sum of all positive integers less than 300 which are divisible by 3.

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Q: How many numbers between 1 and 10000 have the digits 1,3, and 5 such that each digit appears exactly…

A: Here Numbers between 1 and 10,000 have digits 1,3,5 and each appears exactly once only

Q: 9. Prove that there are as many positive real numbers as there are real numbers.

A: Consider the set X and Y. Then we said that |X| = |Y| if there exists bijective function from X to…

Q: If n is an even integer then 7- n3 ) is an odd integer. ) is an even integer. is sometimes even and…

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Q: A positive integer n is called superperfect if ơ(ơ(n) = 2n Show that 16 is superperfect.

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Q: noosing a positive integer from 100 to 999, what is the poSSibility of choosing a positive integer…

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Q: How many positive integers between 50 and 100 a) are divisible by 7? Which integers are these? b)…

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Q: 32. Prove that 3 divides n + 2n whenever n is a positive integer.

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Q: "Every positive integer can be written as the sum of the square of 3 positive integers." prove…

A: Given: "Every positive integer can be written as the sum of the square of 3 positive integers." To…

Q: The sum of all positive integers less than 300 which are not divisible by 3.

A: Sum of n positive integer is given by, Sn=nn+12

Q: Show that there are infinitely many prime numbers.  0 /∈ N. (zero is not an element of natural…

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Q: Find the least positive integer n for which 2" + 5" - n is a multiple of 1000.

A: To find the value of n substitute n=1,2,3... multiple of 1000 is a certain number divided by 1000…

Q: If n > 1 is an integer not of the form 6k + 3, prove that n + 2" is composite. [Hint: Show that…

A: Congruence modulo relation on integers: Let a, b∈Z and let m be a fixed positive integer. Then we…

Q: 5. Given a real number x and a positive integer k, how many multiplications are used to find kx by…

A: Given,A real number x and positive integer kTo find: Total number of multiplications used with the…

Q: (a) Suppose that there are k distinct positive integers that divide an odd positive integer n.…

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Q: Prove that 4 divides 6 7" - 2.3" for all integers n 20.

A: We need to prove that 4 divides 6.7n-2.3n for all integers n≥0.

Q: 3.) An airplane takes off and then climbs 2600 feet. After 25 minutes, it descends 150 feet. What is…

A: Please note that as per our guidelines if multiple questions are posted, we can answer the first…

Q: Can you please explain to me why is (k(k+1) ) / 2 is an integer and I know it always even so can you…

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Q: If n is a positive integer, is there always a prime between n and 2n?

A: given that n is a positive integer. we have to show that there is always a prime number between n…

Q: The sum of all positive integers less than 400 which are divisible by 5.

A: Positive integers less than 400 which are divisible by 5 are⇒n=79 5,10,15,20,...,395 The above…

Q: Use a direct proof to show that the sum of two odd integers(k and l) is even.

A: The number n is even if it is expressed as  n=2m , where m is an integer. The number n is odd if it…

Q: 9. Let n be an odd number with exactly four distinct prime factors. If (3, n) = 1, prove that n is…

A: We will use knowledge of number theory to answer this question.

Q: how many positive integers n satisfy 22-5n>-3?

A: 22-5n>-3

Q: Choose two distinct integers, x and y, between 0 and 9. Derive the number of 9- digit integers that…

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Q: "For all integers n, n(n² – 1) is divisible by 6." (You may use quotient-remainder theorem and…

A: Let us consider, for all integers n a = n3 – n a = n (n2 – 1) a = n (n + 1)(n – 1)

Q: How can the sum of k * (n choose k) = n(2 to the power of n-1) be proven akgebraically ?

A: Step 1: We have to prove that, Now by the binomial theorem, Now, differentiating the both sides of…

Q: 4. A property that, two consecutive positive integers are relatively prime. Consider any two…

A: Two consecutive numbers are said to be relatively prime or coprime numbers if they have no common…

Q: The number of primes of the form n³ – 1 is 1) 1 2) 2 3) Infinite

A: To choose correct option

Q: Can you think of the smallest number N such that ot can be expressed as a product of the two factors…

A: We have to find out the number N that can be expressed as a product of the two factors which are…

Q: 5) Prove that the sum of two fourth powers of integers can never be three more than a multiple of 4.…

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Q: 3. There are seven students who always compare their CS test grades. They discovered that either the…

A: It is given that there are seven students who always compare their CS test grades. They discovered…

Q: If n is an even integer then 7-n3 is sometimes even and sometimes odd. O is an odd integer is an…

A: We have to state whether the 7-n3 is an even integer, odd integer, or sometimes even and sometimes…

Q: Consider all positive integers with seven different digits. (Note that zero cannot be the first…

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Q: Give a proof by contradiction that there does not exist an integer which is both even and odd.…

A: Assume an integer x which is both odd and even.

Q: Let ab5c be all four-digit multiples of number 36 with different digits. If someone found 7 numbers…

A: Let ab5c be all four digit multiples number 36 with different digits. As 36= 4x9, so the number ab5c…

Q: Consider all positive integers with seven different digits. (Note that zero cannot be the first…

A: Given that we have to find all positive integers with seven different digits which satisfy the given…

Q: What 2 nonnegative numbers with a sum of 23 have the largest possible product?

A: Given information: The two non negative numbers with a sum of 23.  Calculation: Consider x and y are…

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.