Use the Principle of Mathematical Induction to verify that, for n any positive integer, 6^n −1 is divisible by 5.

Q: Use mathematical induction to show that 5? − 1 is divisible by 4. Determine the starting value as…

A:

Q: Suppose you draw n lines in the plane so that every pair of lines cross (no lines are parallel) and…

A: Let n be the number of non parallel lines. Given that number of regions= Rn Let Un represent maximum…

Q: There are 8 people in a room, whose ages are between 10 and 40. Show that we can always find two…

A: Pigeonhole principle Here, we don't know the ages of 8 people, but we know the range of sum of their…

Q: Show that 8(-n--8()

A: By properties we know that,

Q: In the expansion of + 6)-", state the number of terms and the coefficient or the middle term. 21, 20…

A: Introduction: The binomial expansion is a very important finite series in mathematics. With the aid…

Q: Pigeonhole Principle Prove that among any 100 integers there always two whose difference is…

A: The given problem is to prove the following statement which follows the pigeonhole principle, we…

Q: Rationalize the denomi 20 5/10 - 7/5

A: 20510-75

Q: Find the number of positive integers X smaller than 12849 which have exactly 2022 positive divisors

A: Theory : Total no of divisor of a number, If its prime factorization, given by ∏i=1hpiai Where p is…

Q: Determine the number of integers between 1 and 10000 that are not divisible by 6, but are divisible…

A: Given, the number of integers between 1 and 10000 To find the integers between 1 and 10000 that are…

Q: Show that a number with 30 digits cannot have more than 100 prime factors.  

A: Prime factor: A factor of a number which is a prime number. For example: consider a number 26. The…

Q: A sequence of positive integers starts with one 1, followed by two 2s, three 3s, and so on. (Each…

A: Given:     The sequence is:     1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,... To get this number, we have to…

Q: A number N is divisible by 4 iff the number formed by the last 2 digits of N is divisible by 4. For…

A: We can find number divisible by 4  by using the concept given in the question

Q: If 6th  term of GP is 9375 and its 4th term is 375.Find its 9th term

A:

Q: Show that 3 + V5 is inational number

A: Let 5^1/3 is a rational numbers. So, we can write it as 5^1/3=p/q ………….(I) according to the…

Q: At the end of 2005, the largest known prime number was 2^30402457 -1, write an expression for the…

A:

Q: How many 5-digit even numbers can be formed using the digits 2, 4, 6, 7, 9 if no repetition is…

A: Given:

Q: Suppose 51 cars start at a car race. In how many ways can the top 3 cars finish the race? The number…

A: Solution: From the given information, number of cars start at a car race is 51.

Q: All composite number can represent as product of primes numbers

A:

Q: Compute the sum of all integers between 60 and 150 that are exactly divisible by 8.

A: Consider the given range of number 60 to 150

Q: Explain how to expand (a + b)' using the binomial theorem n an-kbk. Be specific. k expression, k-0

A: Given:      a+b7 The binomial theorem:        a+bN=∑n=0NNnaN-nbn where Nn=N!n!N-n!

Q: What is the 5th term in the expansion of

A: The 5th term of the given expansion is solved by using binomial expansion formula (X+Y)^n.

Q: Determine the smallest integer that can be expressed as a sum of two primes in three different ways

A: Let us have smallest prime numbers  are, 2,3,5,7,11,13 17,19.....

Q: how to prove √6 is irrational

A:

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

A:

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

A:

Q: Show that, selecting 91 positive integers guarantees that there are at least 10 integers with the…

A:

Q: Find the smallest integer greater than 100 that is divisible by 3 but leaves the remainder 1, 5 when…

A: To find- Find the smallest integer greater than 100 that is divisible by 3 but leaves the remainder…

Q: Determine the probability that the randomly selected number from the interval [0,1] is a rational…

A: To Determine: The probability that the randomly selected number from the interval [0,1] is a…

Q: Find the sum of those integers between 1 and 500 which are multiples of 2 as well as of 5

A:

Q: A hot air balloon descends at a rate of 4ft per minute for 6 minutes, then rises at a rate of 2ft…

A: Given Data: The descending rate of the balloon is: D'=-4 ft per minute The time till the balloon…

Q: Sort this permutation [5 213 4] into identity permutation [123 4 5]. Check what's the number number…

A: The permutation is nothing but the arrangement of objects in different orders. If we permute a set…

Q: Describe a general method for finding all the divisors of a positive integer once you know the prime…

A: Given  A positive Integer whose prime factorization is given . To find all the divisors with general…

Q: Prove that if the sum of the digits is subtracted from a natural number, then the result is…

A: let us assume the number to be three digit number xyz therefore, xyz=100x+10y+z sum of digits will…

Q: Integers with different values in each digit can be expanded in a similar way. For these integers,…

A: Let we have a number 5267843 This number can be expanded as…

Q: In two or more complete sentences, explain how to find the third term of the expansion of (2x+y)4.

A:

Q: what is the smallest positive integer greater than 1 that divides the number 7^2021-13^2021

A: here 7×2021-13×2021  =2021(7-13)                                       =2021(-6)…

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: Using the fast exponentiation method (page 182), determine 535 mod 47. Show your work.

A: We will find out the required value.

Q: (2) Explain why the rational numbers are those having periodic representation or with- out infinite…

A:

Q: A DIGITAL CHIP MADE BY YOUR. MICROELECTRONICS DIVISION, COUNTS CONTINUOUSLY IN THE SEQUENCE: 1024…

A: The counts of the chip are 1024,2048,4096,8192,.... Now, we can see that 20481024=40962048=2 Each…

Q: Fermat's theorem is true for only prime number. إختر واحداً.

A:

Q: How many twin primes are there where both primes are less than or equal to 31? Number of twin…

A: Twin prime: If there is only one composite number between two prime numbers, then that two numbers…

Q: Find the number of positive integers ≤ 1000 that are not divisible by 5, 13, or 31?

A: .  Symbol of Greatest Integer Function  To Find: Number of positive integers ≤ 1000 that are not…

Q: How many natural number that are less or equal to 1000 and that are not multiplea of 3,5or7 exist

A: To find: Number of natural numbers ≤1000 that are not multiple of 3 or 5 or 7.

Q: 5 divides n -n whenever

A:

Q: Without expanding completely, find the indicated term(s) in the expansion of the expression.

A: Given expression: 3c+c247 Concept used for the general term:…

Q: Show that 19 divides 10*8n-1 + 33n-1 for every positive integer n.

A: Concept: Mathematical induction: For each and every natural number n, mathematical induction is a…

Q: algorithm to determine the decimal number answer to 2893 ÷ 6 to the hundredths place. b. Interpret…

A: Solution of the problem as follows

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…