5. Show that 2 32 3" 31 3" for all positive integers n.

Q: 3. Prove the statement "6 divides n³ - n for all integers n> 0".

A: We need to prove that 6 divides n3-n for all integers n≥0.

Q: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

A: We have to find how many 5’s does the prime factorization of x + y contain.

Q: 2. (a) Find integers a, m, n such that m | a and n | a, but mn /a.

A: .

Q: If n is a positive integer with (n + 1)(n + 3) is odd, then (n + 2)(n+ 4) must be a multiple of 1) 6…

A: Given:  (n+1)(n+3) is odd We will get an odd number from product of two numbers only if the two…

Q: Prove that 3"< n! whenever n is a positive integer greater than 6.

A: Here we are to prove 3n&lt;n ! whenever n is a positiveinteger greater than 6.To solve this problem…

Q: Prove: For integers m, n, 12mn – 9 is an odd integer.

A:

Q: Show that for positive integer n: (2n)! VT. 22n . n! Г 2

A:

Q: 7. Suppose that m, n E N are two consecutive numbers such that both m > n > 2. Prove that if n is a…

A:

Q: 3. Let n and m be positive integers. Evaluate > (km+1 (k – 1)m+1) if possible. If not possible, k=1…

A: We have to evaluate the given summation.

Q: V. Show for all integers n2 1 that 71(3*n+ – 52n-). а.

A: As you have asked more than one question from different backgrounds, I am solving the first question…

Q: (2) Show that for any integers a,b we always have a? – 4b # 2. -

A:

Q: Show that2/31 + 2/32 + g+ 2/3n = 1 - 1/3nfor all positive integers n.

A: The given series is the geometric series, hence the formula to find the sum of the series helps to…

Q: п For all integers n > 1, 1.2 2.3 п(п+ 1) п+ 1

A:

Q: 2. Suppose a, b, c, d are integers and that (a, b) = (c,d) = 1. If (g) + (á) show that b = ±d. is an…

A:

Q: Suppose thatk and n 2k are fixed positive integer. Justify the identity k+1 jk

A: to prove ∑j=knjk=n+1k+1above equation can be written as S2(j,k)=S2(j-1,k-1)+k.S2(j-1,k)count the…

Q: b) Prove that () - 2) = n², where n is a positive integer.

A:

Q: 1 ιπ 1s +472, e where n is an integer.

A:

Q: 7. For every integer n z 1, n(5n – 3) 1+6+11+16++(5n– 4) 2

A: Solution is in step 2

Q: or disprove : [nx] =n [x] %3D for all Positive integers n > 2

A:

Q: Suppose that k and n > k are fixed positive integers. Justify the identity k+1,

A: Given k and n≥k are fixed positive integers.

Q: 2. Possible whenn A- Zero b. Negutive integer C- Negative Yational d- Positive integer

A: We know that Laplace transform of

Q: that for any integer n, n -n is even.

A: Prove that for any integer n, n3-n is even.

Q: Prove the statement for positive integer n and r, with r ≤ n. P(n, 1) = n

A: To prove the statement

Q: Show that if n is a positive integer, then (n + 1, n2 - n + 1) = 1 or 3

A:

Q: 4. Let n and k be two positive integers with n 2. Prove that (n 1)² divides n-1 if and only if n -1…

A: The given problem is to prove the given following statement. Given statement, let n and k be two…

Q: 6. Prove that 3" < n! whenever n is a positive integer greater than 6.

A: We need to prove that ,  3n &lt; n!  for every positive integer n greater than 6. We will use method…

Q: 4. Argue by contradiction that for all integers n > 2, there is a prime number p such that n < p<…

A: Fact-1: If n is a positive integer and n is greater than or equal to 2, then n must have a prime…

Q: Show that if n is a positive integer, then k+4 n(3n +7) k(k +1)(k+2)¯ 2(n+1)(n+2) k=1

A:

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

A:

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: 2. Determine all values of x - y when x and y are integers such that x = 360 + y².

A: The given equation is. x2=360+y2x2-y2=360x+yx-y=360 Since it is given that x and y are integers, so…

Q: 2. Show that if a is an integer other than 0, then a) 1 divides a h) a divides 0

A:

Q: Prove that if 10|n, then 10|n^2+n

A: Prove that if 10|n, then 10|n^2+n

Q: If nis a positive integer 1 then prove (D - a) n!

A:

Q: Disprove the claim: 3 m E N such that m 2 3 and m? – 1 is a prime number. -

A: Given that ∃m∈ℕ such that m≥3 and m2-1 is a prime number. Take m=4. It is known that 4∈N And 4&gt;3.

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: Show that for any integer n ≥ 2, C(2n,n) > C(2n,n-1)

A:

Q: For any positive integer n and r = 1,2, ...,n, we have G) = (","') + C" - }). True O False

A: The expression nr can be presented in the factorial form as follows. nr=n!(n-r)!r! The expression…

Q: Check that o(-2)" (":) = 1– 2(;) + 4(;) + ... + (-2)" (") + ... + (-2)" in ir=0 cannot be greater…

A: Given sum is  ∑r=0n-2rnr=1-2n1+4n2+...+-2rnr+..+-2nWe need to prove that the given sum cannot be…

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: Prove that n² + 1> 2" when n is a positive integer with 1<n<4.

A: Given: is a positive integer with. To prove: .

Q: Hello, any help for this problem would be great, thank you!

A:

Q: Prove: 5" +9 < 6" for all integers n2 2.

A:

Q: 9. Show that if n is a positive integer then n Σ k² ( x ) = n(n + 1)2²−² k k=0

A: Concept: Mathematical induction: Mathematical induction is a technique for proving that a statement,…

Q: Show that for all n € N, one has 1 2√n+1 <√n+1-√n<; 2√ñ

A:

Q: Let x and y be positive integers. When is x + y3 a prime number? When is x+y a composite number?…

A:

Q: Show that if n is a positive integer then n(n +1)2"-2 k=0

A:

Q: Prove or disprove the following: n+1 For any odd integer n, H= ()) 2

A: Let  n=2k+1         -  (1) Where k is an integer. Then n is any odd integer. Now, LHS= n24        -…

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.