3. Show that 13" – 7ª is divisible by 6 for n > 1.

Q: Let n ∈ Z. Prove that if n is divisible by 3, then n = 6k or n = 6k + 3 for some integer k.

A:

Q: 15) n(n*+5) is divisible by for each integer

A:

Q: 1. Prove that 7" -1 is divisible by 6 for all n > 1.

A: As per Bartleby Guidelines we can solve only one question at a time. Hope you understand

Q: 2. Find the missing numbers in the magic square below. 2 13 10 11 6. 12 15 1 4-

A: Consider the provided table,

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 n! > 3ª for n> 7.

A:

Q: Show by induction that 19"-8" is divisible by 11 for all n >0

A:

Q: 3. Show that ifn = 2 (mod 4), then 9" + 8" is divisible by 5.

A: Here we will prove the given statement.

Q: Prove that (n – 1) +n° + (n + 1)° is divisible by 9 for n E {1,2,3, . }

A:

Q: 9.) Is 9" + 3 is divisible by 4? PROVE IT!

A: Given : 9n+3 is divisible by 4. To Prove: The given statement is true for n=k+1. Consider Tn=9n+3.…

Q: 4. n(n+ 1)(n + 2) is divisible by 6

A: We have to check

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: Number Theory

A: Given that n is a positive integer. The objective is to prove that

Q: (i) Deduce that the difference in the number of integers in the numerator and denominator of (* 2N N…

A: Properties of floor functions will be used.

Q: Vn E N, nº – n is divisible by 6.

A:

Q: Find the order of [5] in Z23

A:

Q: Use Fermat's Theorem to show that n 13 - n is divisible by 2730 for all n.

A:

Q: 5. Prove that the integer 3". 22" – 1 is divisible by 11 for all n > 0.

A:

Q: 3. Show that for all integers n: a) if d is an integer such that d |n + 8 and d | n² + 1, then d |…

A: To solve the given questions we will use the following formulas : 1.   We say that two positive…

Q: (b) An integer is divisible by 3 if and only if the sum of its digits is divisible by 3.

A:

Q: Suppose that n=24.35.112.133. How many divisors does n have?

A: Given,  n = 24 35 112 133 Then (4+1)(5+1)(2+1)(3+1) = 5×6×3×4=360 It has 360 number of divisors

Q: Compute. 13 2) Σ k

A: Given, ∑k=113k2

Q: Suppose m = apɑp-1*** a2ɑ1ao and n = bpbp-1*…* b½b¡bo are two numbers with the same digits so that n…

A: 7) a. Given that ai=bj. Without loss of generality assume that i>j. Now the value of the digit ai…

Q: 2. Use induction to prove that 9" – 1 is divisible by 8, if n E N!

A: We need to use induction to prove 9n-1 is divisible by 8 . Now , we use induction hypothesis to…

Q: 2. Prove that n(n + 1)(n + 2) is always divisible by 6

A: We have to show that the number nn+1n+2 is always divisible by 6 for all n∈ℕ. Let Pn=nn+1n+2 where…

Q: 28. Suppose that [E:Q] = 2. Show that there is an integer d such that E = Q(Vd) where d is not…

A: Given [E:D]=2.

Q: [Q1] From the letters {A, B, C, D, E, F, G}, how many: (a) 3-distinct-letters co can be generated?…

A: Given: There are 7 letters A, B, C, D, E, F, G

Q: 2. Prove by the method of mathematical induction that for any integer n 21 the expression 8" + 13 is…

A: The solution is given as

Q: 1. Show that b, = (n!)'/n > 20.

A: Consider n!1n=an1n.

Q: Find all integers n> 1 such that 1" + 2"+... +(n – 1)" is divisible by n.

A:

Q: Show that 3-4" + 51 is divisible by 3 and 9 for all positive integers n.

A:

Q: (c) Prove that n? + 1 is not divisible by 3 for every integer п.

A:

Q: 16. Determine the value of n in simplest form: 1" +i"

A:

Q: (a) Prove that 82n – 12" is divisible by 13 for all nɛ N.

A: As per guidelines we are supposed to solve maximum one questions at a time.kindly post another…

Q: Show that in Zm there is no nilpotent element except 0 by letting m be the product of three distant…

A: Statement : In Zm, there is no nilpotent element except 0 We need to prove the  above statement by…

Q: How many terms are there in the expansion of (a+4b)^20 ?

A: Given expression, (a+4b)20

Q: n that includes all terms up to

A: The initial value problem is , w''+7xw'-w=0. It is given that w0=6, w'0=0. w''+7xw'-w=0. To find the…

Q: 2. 6n – 1 is divisible by 5 for n any positive integer

A:

Q: If n and a are prime to 7 then prove that nº – a® divisible by 7?

A:

Q: Show by induction that 21" – 8" is divisible by 13 for all n> 0

A: For the solution follow the next steps.

Q: Is it true that for any odd n the number 2" +1 is divisible by 3? Prove your answer.

A: We prove this by using PMI(Principle of Mathematical induction)

Q: 1. Prove: 11" – 6 is divisible by 5 for all positive integers, n.

A:

Q: The greatest common divisor of 9n + 10 and 10n +9 is Oa 10 or 19 оь. 19 O. None d. 1 or 19 Oe. 1

A:

Q: nP 00 クの In=1 2+n3

A:

Q: Show by induction that 19" – 8" is divisible by 11 for all n >0

A:

Q: For all n>= 1, then P(n) = n-n is divisible by 3 Select one: O True O False

A:

Q: Prove that n(n + 1)(n + 2) is always divisible by 6

A:

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

A: