menu
bartleby
search
close search
Hit Return to see all results

Claim: For any natural number n, if 6 ∣ n, then 3 ∣ n.

Question

Claim: For any natural number n, if 6 ∣ n, then 3 ∣ n.

fullscreen
check_circleAnswer
Step 1

Use the mathematical induction to show that the given statement For any natural number n, if 6  n, then 3  n” is true for all natural numbers n.

 

Let the number 6 can be written in the form n(n+1)(n+2) because n(n+1) (n+2) is always divisible by 6.

 

Using mathematical induction:

 

for 1 the value:
n(n+1)(n+2)1(2) (3) = 6 = 2 . 3
similarly for 2 the value:
n(n1)(n+2)-2(2+1)(2+2) = 6.4=2.3-4
both the above value is divisible by 3
help_outline

Image Transcriptionclose

for 1 the value: n(n+1)(n+2)1(2) (3) = 6 = 2 . 3 similarly for 2 the value: n(n1)(n+2)-2(2+1)(2+2) = 6.4=2.3-4 both the above value is divisible by 3

fullscreen
Step 2

let the statement is true for n = k so k(k+1)(k+2) is divisible by 3.

 

Then the statement n(n+1) (n+2) is always divisible by 6 is true for n = k+1.

 

So,

n(n+1) (n2) for n= k+1
k+1(k1+1k 1+2) =(k+3)(k +3k+2)
= (k +3k +2k)(3k +9k+6) or k(k2 +3k+2)+3(k +3k+ 2)
help_outline

Image Transcriptionclose

n(n+1) (n2) for n= k+1 k+1(k1+1k 1+2) =(k+3)(k +3k+2) = (k +3k +2k)(3k +9k+6) or k(k2 +3k+2)+3(k +3k+ 2)

fullscreen
Step 3

So, the above statement is true as the above expression must be divisible by 6. And we have to show that the same expression is divisible by 3. The expression 3(k2+3k+2) is always divisible by 3 and the other expression k(k2+3k+2) = k(k+1)(k+2)

 

So,

...
6n k(k +1)(k+2)+3(k +1)(k+ 2)
3+3n,(k1)(k + 2) + 3(k +1) (k+ 2)
so.
3,k(k1)(k2)
help_outline

Image Transcriptionclose

6n k(k +1)(k+2)+3(k +1)(k+ 2) 3+3n,(k1)(k + 2) + 3(k +1) (k+ 2) so. 3,k(k1)(k2)

fullscreen

Want to see the full answer?

See Solution

Check out a sample Q&A here.

Want to see this answer and more?

Our solutions are written by experts, many with advanced degrees, and available 24/7

See Solution
Tagged in

Math

Advanced Math

Related Advanced Math Q&A

Find answers to questions asked by student like you

Show more Q&A add
question_answer

Q: Let G be a finite group. Let xeG, and let i>0. Then prove that o(x) gcd(i,0(x))

A: To prove the required identity on the order of the group element

question_answer

Q: Theorem 4P.1 There are exactly six lines in the four-point geometry.   proof the Theorem 4P.1 by con...

A: To prove that in the Four point geometry , there are exactly six lines: the proof should be based on...

question_answer

Q: If b is dviisible by a (a|b) and b in different than 0 then |a| is less or equal than |b|

A: The given statement means the following.

question_answer

Q: Assume that you have a total of 9 people on the board: 3 out-of-state seniors, 1 in-state senior, 2 ...

A: To work out the number of selections under the given conditions

question_answer

Q: What is the probability of a paint defect What is the probability of a paint defect which includes a...

A: (1) Obtain the probability of a paint defect.

question_answer

Q: what is the product of 9/11 and it's reciprocal

A: The given fraction is 9/11.The reciprocal of the number is 1 / (9/11).That is, 11/9.

question_answer

Q: 2*. Let Q/Z be the group described in problem 12 of Worksheet 1.1. Find list the elements of the sub...

A: To identify the subgroups generated by the given elements in the quotient group Q/Z.

question_answer

Q: 3-B. Write each answer with a reasonable number of figures. Find the absolute and percent relative u...

A: (a)Given that [12.41(±0.09) / 4.16 (± 0.01)] × 7.0682 (± 0.0004).

question_answer

Q: In the following problems, decide if the groups G and G are isomorphic. If they are not, give proper...

A: (a)   We are given that G = GL(2, R), the group of 2 × 2 non-singular matrices under multiplication;...

Sorry about that. What wasn’t helpful?