NUMBER THEORY: Show that 8^900 - 7 is divisable by 29 Help

In summary, number theory is a branch of mathematics that deals with the properties and relationships of numbers, particularly integers. To show that a number is divisible by another number, one can use the division algorithm or perform long division. To prove that 8^900 - 7 is divisible by 29, one can use the fact that any number raised to a power that is divisible by 28 will have a remainder of 1 when divided by 29. Another method to prove this is by using modular arithmetic. It is important to prove this divisibility because it establishes the validity of the result and provides a deeper understanding of numbers and their relationships.
  • #1
tamintl
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0
NUMBER THEORY: Show that 8^900 - 7 is divisable by 29 ... Help

Homework Statement


Show that 8^900 - 7 is divisable by 29


Homework Equations





The Attempt at a Solution



By Fermats little theorem

(8^28)^32 x 8^4 - 7
=1^32 x 8^4 - 7
=8^4 - 7
=(8^2)^2 - 7
=64^2 - 7

NB: 64 = 29x2 + 6

therefore: 64 => 6

=(6)^2 - 7
= 36-7

29=0 mod 29



Is this correct or is there a quicker way?

thanks tamintl
 
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  • #2


That is correct and it's the quickest way I can think of to show it.
 

Related to NUMBER THEORY: Show that 8^900 - 7 is divisable by 29 Help

1. What is number theory?

Number theory is a branch of mathematics that deals with the properties and relationships of numbers, particularly integers.

2. How do you show that a number is divisible by another number?

To show that a number is divisible by another number, you can use the division algorithm or perform long division to see if the remainder is 0.

3. How do you prove that 8^900 - 7 is divisible by 29?

To prove that 8^900 - 7 is divisible by 29, we can use the fact that any number raised to a power that is divisible by 28 (29-1) will have a remainder of 1 when divided by 29. Therefore, we can rewrite 8^900 as (8^28)^32, which is equivalent to 1^32. Thus, 8^900 - 7 is equivalent to 1 - 7 = -6. Since -6 is also divisible by 29, this proves that 8^900 - 7 is indeed divisible by 29.

4. Can you use a different method to show that 8^900 - 7 is divisible by 29?

Yes, another method to show that 8^900 - 7 is divisible by 29 is to use modular arithmetic. We can rewrite 8^900 as (8^2)^450, which is equivalent to 64^450. Since 64 is congruent to 6 (mod 29), we can write (64^450 - 7) as (6^450 - 7) and use the fact that any number raised to a power that is divisible by 28 (29-1) will have a remainder of 1 when divided by 29. Therefore, (6^450 - 7) is equivalent to (1 - 7) = -6, which is divisible by 29.

5. Why is it important to prove that 8^900 - 7 is divisible by 29?

It is important to prove that 8^900 - 7 is divisible by 29 because it helps to establish the validity of the result and provides a deeper understanding of the properties of numbers and their relationships. This proof can also be used in other mathematical computations and can serve as a basis for further research in number theory.

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