Use the ertended pigeonhole principle to show that there is at least 15 ways to choose 4 integers from 1 to 12 so that all the choices have the same sum.

Q: Find the least amount of different numbers to pick from positive integers that are at most 100 to…

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Q: How many integers between 100 and 400 contain the digot 2?

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Q: Select two positive integers a and b with 5 <  a  ≤  20  and 5  <  b ≤  20.

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Q: Suppose that n pigeons are placed into m holes with m ≥ k(n+ 1) for some positive integer k. Suppose…

A: See the attachment

Q: How many positive integers less than 10000 are there which contain at least one 7 or at least one 9…

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Q: Produce the least positive number that is divisible by​ 2, 3,​ 4, 5,​ 6, 7,​ 8, 9,​ 10, and 11.…

A: Given Produce the least positive number that is divisible 2, 3,​ 4, 5,​ 6, 7,​ 8, 9,​ 10, 11.…

Q: How many six-digit positive integers contain exactly one one, two twos and three treor? How many…

A: First note that here six digit integer can start only with 1 or 2. If it starts with 1 the remaining…

Q: Assume an unlimited supply of coins worth 5 cents and 9 cents. Prove that some

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Q: In a set of 2 positive integers, how many ways it can be chosen in less than 50? 1.

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Q: Show that if n is any integer, then 3 divides n3 − n

A: To prove:  if n is any integer, then 3 divides n3 − n. Concept used: Here we shall use the identity:…

Q: Show that an integer N is divisible by 9 if and only if the sum of its digits is divisible by 9

A: Let a=anan-1...a3a2a1 be an integer and S be the sum of the digits. Consider a,…

Q: How many positive integers less than 1,000,000 have a digit sum of 5?

A: We have to find the number of positive integers 1,000,000 have a digit sum of 5. The number less…

Q: Texas is starting a lottery game. Play- ers use a machine that randomly selects 7 distinct numbers…

A: To calculate the probability that Texas will lose money, we need to determine the expected value of…

Q: Determine whether the statement is true or false, "3 is an even integer and 3 is less than 5."

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Q: How many triples of integers from 2 through 16 can be formed in which the integers are in increasing…

A: Any triple of integers (i, j, k) with 2 ≤ i ≤ j ≤ k ≤ 16 can berepresented as a string of 15 – 1…

Q: How many 5–digit (decimal) integers are there with none of the digits repeated?

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Q: Derive an expression for the sum of the first n positive even integers.

A: We Use the A.P series. Find The sum of the first n positive even integer.

Q: Prove that the sum of any five consecutive positive integers is always a multiple of five.

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Q: Prove that a three-digit number is a multiple of 9 whenever the sum of the digits is a multiple of…

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Q: Jse a direct proof to show that if n² is an odd integer, then n* is also an odd integer.

A: Given, n² is an odd integer. So, n² = 2k+1, where k is an integer. Now, n⁴ = (n²)² = (2k+1)² = 4k² +…

Q: Find the sum of the positive even integers up to and including 50.

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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: "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: Prove the statement that "The sum of three consecutive integer numbers is odd if and only if the…

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Q: Prove a number is divisible by 3 if and only if the sum of its digits is divisible by 3.

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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: Find three even numbers less than 400, each of which can be expressed as asum of consecutive…

A: The objective is to obtain 3 even numbers less than 400 such that they can be expressed as a sum of…

Q: Solve the separable equation (2/1+t²) dy = (ty¬l)dt subject to y(0) = -Va where a > 0 is a constant.…

A: The given equation is first order separable differential equation:

Q: An n×n magic square is a square array of numbers (n > 1) such that the sum of every row, of every…

A: Given a n×n magic square

Q: Prove: “The sum of two even integers is even.”

A: To prove: The sum of two even integers is even. n is an even number if it can be expressed as n =…

Q: Find the sum of the first 20 positive even integers.

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Q: John found the sum of the first 2021 positive even integers and Bob found the sum of the first 2022…

A: In this question, tne concept of sum of first positive even integers is applied.