that if 3 divides the sum of the digits of n, then 3|n

Q: d) How many 4 digit numbers can you make from the elements of A without repetition contain the…

A: We shall solve first question only as per answering guidelines. If you want others to be solved…

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: Find the greatest natural number n such that n + 10 divides nº + 100.

A:

Q: show that n(n^2-1)(n+2) is divisible by 4 for all integers n

A: To show that n(n^2-1)(n+2) is divisible by 4 for all integers n

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: If n is an integer and n 2 1, then the expression 4" + 4-1.3+42.3+ 4-.3+4".3++4 3+ 3 is equal to

A: (c) is correct We have to find the value of 4n+4n-1·3+4n-2·3+4n-3·3+4n-4·3+⋯+4·3+3

Q: For how many values of n is the sum 1 + 2 + · • + n not divisible by n if 1 ≤ n < 100?

A:

Q: Show that if m is an integer then 3 divides m3−m.

A:

Q: How many integers n where 1≤n≤1600 satisfy the condition ⌊n⌋\n.

A: Here, need n such that n/⌊n⌋ where 1≤n≤1600 and n is integer. Now, ⌊n⌋ is said to be floor function…

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: If k is a positive integer, find the (n!)k + 6 Σ ((k + 6)n)! n = 0 R =

A:

Q: Find all the odd positive integers n which satisfy 6(n) = 20.

A:

Q: Show that if n is a positive integer then =n(n + 1)2"-2

A: We know the binomial theorem formula isx+yn=∑k=0nCkn xn-kyk, where Ckn is the binomial co-efficient…

Q: What is the smallest integer n for which 25n&gt;512?

A: The provided expression is as follow:

Q: How many three-digit numbers can be formed under each condition? (a) None of the digits can be 5. X…

A: when there are no condition applied we have 10 choices for each digit : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Q: 6.) Find the sum of the first 50 positive integers ending in 6.

A:

Q: (b) Find the sum of the first 110 positive odd integers.

A: Our Aim is to find the sum of the first 110 positive odd integers.

Q: If n = p,P2 ... P where the p, are distinct prime numbers, how many ways can n be written as a…

A: First let us formulate the problem in a different way. Look! we have to find the total number of…

Q: 1. Find the sum of the first 50 even positive integers.

A:

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: Let n = 9. Find n - 1. n - 1 = a/5

A:

Q: How many integer solutions to x1 + x2 + x3 + x4 = 32 are there for which x1≥6, x2 ≥5, x3 ≥3 and x4…

A:

Q: If in the expansion of (1 + n}+", the 5th term is 24 times the 3rd term, fnd the value of n.

A:

Q: For n ≥ 2, let an=5n/n2-2 Determine for which n, |an|<1

A: Use the given inequality for an and find the quadratic equation for n and solve

Q: Show by contradiction that for any integer n: if n>0, then n+10 n+5

A:

Q: Show that 2^n &gt;n^2 if n is large enough.

A:

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: Show that the sum of the squares of the first n natural numbers is given by,S=(n(n+1)(2n+1))/6

A: We have to show that the sum of the squares of the first n natural numbers is given by S=nn+12n+16…

Q: Show that n! > 3 for n ≥ 7

A: We have to prove this assertion that, n! &gt; 3  for n≥7 We will prove this statement by the method…

Q: is an integer greater than 1 Quantity a= the tens digit of N Quantity b= 0 which is greater or…

A:

Q: For n≥2, let an= 5n/n2-2. Determine for which n, |an|<1.

A: Use the inequality on and find the inequality for n

Q: Prove Γ (n) = (n-1)! where n ∈ ℕ

A: Concept: The proof for this gamma integral will be done using mathematical induction

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

A:

Q: The sum of the digits in the number 56 is 11. (5+6=11) a) Show that the sum of the digits of the…

A: a) Sum of the digits from 15 to 18 1+5=6 1+6=7 1+7=8 1+8=9 Now 6+7+8+9=30

Q: Prove that 2^? +3^? is a multiple of 5 for all odd ?∈N.

A: We will solve this problem using principle of mathematical induction.

Q: Show that 5 | n5-n for all integers n.

A: To prove the above statement we will use the principle of mathematical induction which is divided in…

Q: For each natural number n, if 3 does not divide (n² + 2), then n is not a prime number or n = 3.

A:

Q: Show that 2^n &gt;=1/8 for n&gt;= -3.

A:

Q: Show that every odd integer is the difference between two consecutive perfect squares.

A:

Q: If n is 0 what is 8n

A:

Q: Show that if n is an integer, thenn= [n/2] + [n/2].

A: Floor function x: x gives as output the greatest integer less than or equal to x. Ceiling function…

Q: Find the sum of all natural numbers n such that 2021-n/99 is a natural number ?

A:

Q: How many integer solutions are there to 1 +X2+X3+X4+Xs=31 with

A: Given, x1+x2+x3+x4+x5=31 Number of Solutions when xi≥0 and xi&gt;0.

Q: Let A = (n:n is a multiple of 5 from 5 to 1000} B = {n: n is a multiple of 3 from 3 to 999}

A: The set A contains the multiple of 5 from 5 to 1000, that is A = {5, 10, 15, 20.....1000}.  The set…

Q: The product of all the natural numbers from 1 to n is written as n! it is read as "n factorial" true…

A:

Q: ove that n’ +5n is divisible by 6 for all n E N.

A:

Q: if n zo is an integer, then n!=n(n-)3.21 ...

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.