6. Find the remainder when 721 divided by 43 7. For a given integer n, prove that 3 k-0

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

A:

Q: if 3 does not divide n then 3 divides 2n^2+1

A: Let us Assumue 3 divides (2n2+1) n2+1=3λ Now  2n2+1=n2+1+1⇒(2n2+1)+(-λ)3=1⇒ gcd(2n2+1,3)=1…

Q: Determine whether it is possible to express 1 as a liner combination of 1 = m· 44 + n· 91 ,. for…

A:

Q: 86- The remainder upon the division of x 98 + 4 by x – 4 in Z, is:

A:

Q: What is the smallest positive integer n such that 80325 divides n!

A:

Q: Show that 7 divides (1111)³³³³ + (3333)¹¹¹¹

A:

Q: Prove that 5 does not divide 5n+3 for any integer n.

A:

Q: R.1. Show that among 14 distinct integers, there must be two that have the same remainder when…

A: In order to prove this statement pigeonhole principle has to be applied. According to pigeonhole…

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 if m is an integer then 3 divides m3−m.

A:

Q: Show that if 5 divides n2 with remainder 4, 5 divides 3n3 with remainder 2.

A: Given that 5 divides n2 5 n25 n×n5 n5 n3 As it gives the remainder 4 on dividing by 5. Therefore it…

Q: 2. Determine the highest power of 5 that divides each of the following positive integers. a) 112,250…

A:

Q: 5n. Determine for which n n-2 For nz 2, let on =

A: Here we have to determine the n for which |an|<1 for all n≥2 where an=5nn2-2

Q: How many integers 2 ≤ n ≤ 65 have no prime divisor less than or equal to ¹/³?

A:

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: 5. Find the highest power of 2 that divides each of the following integers. a) (101111110)2 b)…

A: The decimal number is equal to the sum of binary digits (dn) times their power of 2 (2n): decimal =…

Q: If X ~ N(80, 144) then find (90< x<100).

A: GivenX~N(80,144)Mean(μ)=80variance(σ2)=144standard deviation(σ)=144=12

Q: 7 divides n if and only if 7 divides n^2

A: Consider If 7 divides n  To prove 7 divides n2 7 divides n can be written as 7×k=n  Consider 7n2…

Q: Find the total number of integers 'n' such that 2 ≤ n ≤ 2000 and H.C.F. of n and 36 is 1.

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: prove that the one's digit of any integer,n, is the remainder when one divides n by 10.

A:

Q: 2 (a) Work out 14 x 4 – 24 ÷ 2. (b) Find a number between 80 and 90 that is a multiple of both 3 and…

A: We have to solve two subparts in the given problem.                        (a) We need to work out…

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.…

A:

Q: Find the smallest positive interger x such that 1001 x leaves a remainder of 70 when divided by 315

A: The given statement can be expressed as an expression, which ix given below  315y+70=1001x Now,…

Q: Let a be an integer such that the remainder of dividing a by 17 is 15. What is the remainder of…

A: We will find out the required value.

Q: If In a or decimal rounded to at least 4 places. 2, In b = 3, and In c 5, evaluate the following.…

A: Given, lna = 2 lnb = 3 lnc = 5

Q: Find the largest valuc of integer value ofk such that 1 (24 - 1) < 900. 5+7++

A:

Q: a.For the 2019 Comrades marathon, 17268 men and 4357 women entered, 15835 men and 3982 women started…

A: As per bartleby guidelines for more than one question, only 1st one is to be answered. Please upload…

Q: Prove the statement by: a) contraposition b) contradiction

A: Given statement is, “For all integers m and n, if mn is even then m is even or n is even.

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: If n is a perfect number, prove Σ d|n 1 2.

A:

Q: Find the largest negative integer leaves a remainder 2 when divided by 3, 5, and 12. (If it is not…

A: Let us  solve the question in the next step.

Q: Suppose that one selects some subset of integers from the set {1, 2, ., 47). What is the smallest…

A:

Q: 6. What is the sum of all 100 positive odd integers starting with 1? Apply your common sense if it…

A: Given n=100

Q: R.2. Show that among 29 distinct integers, there must be a set of 5 integers all of which have the…

A:

Q: Ifnzo integer with (n+1) (n+3) is ood, then (n+2) (n +9) must be a multiple of 1936 4) None

A: Note- Since you have asked multiple questions, we will solve the first question for you. If you want…

Q: is ​{x | x is a fraction between 71 and 72​} finite or infinite

A: We will solve the problem. There are infinite real numbers between any two natural numbers. These…

Q: 333 111 3: Show that 7 divides 111+ 333

A:

Q: Suppose that p is a prime number greater than 30. Prove that when p is divided by 30, the remainder…

A:

Q: 8. Show that no prime number of the form 4k +3 can divide a number of the form n2 + 1.

A:

Q: Let n be the positive integer n=36!/26! Determine the remainder when n is divided by 13.

A: We will find out the required value.

Q: Find the sum of all integers from 41 to 83. The sum of all integers from 41 to 83 is

A: Detailed Solution and Explanation is given below....

Q: Suppose n is odd. Find the cube of n and divide it by 4. What possible remainders could occur? Check…

A: Solution

Q: 4. Suppose that a and b are integers that divide the integer c. If a and b are relatively prime,…

A: Suppose that a and b are integers that divide the integer 'i'If a and b are relatively prime. Show…

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: 1.What is the smallest number that can be exactly divided by 2, 3, 4, 5  and 8 ? 2.What are the…

A: To answer the given questions.

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.