Solving Integer Cubes & Squares: 7k & 3k?

In summary, the integer a must be both a square and a cube number in order to be simultaneously that is simultaneously a square and a cube number. If a is a square and a cube number, it is a sixth power. The easiest way to solve this problem is using the Fermat's Little Theorem.
  • #1
MAGICMATHS
7
0
I am trying to solve how an integer is simultaneously that is simultaneously a square and a cube number must be either of the form 7k or 7k+1 and I am failing when i work (7k+2)^2, even (7k+3)^2....Can i interpret 7k+1 as 7k+(-1), i think i can't but then i fail in many steps! Also i know that any square number can be expressed as 3k or 3k+1...IS THERE SOMETHING SIMILAR FOR THE CUBIC PLEASE?
 
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  • #2
MAGICMATHS said:
I am trying to solve how an integer is simultaneously that is simultaneously a square and a cube number must be either of the form 7k or 7k+1 and I am failing when i work (7k+2)^2, even (7k+3)^2....Can i interpret 7k+1 as 7k+(-1), i think i can't but then i fail in many steps! Also i know that any square number can be expressed as 3k or 3k+1...IS THERE SOMETHING SIMILAR FOR THE CUBIC PLEASE?

The easiest way is using Fermat's Little Theorem.

Let a be any integer. If a is both a square and a cube, it is a sixth power.

By Fermat's Little Theorem,

[tex]a^7 \equiv a \pmod 7[/tex]

Two cases:

Case 1: [itex]7 \nmid a[/itex]

Hence, [itex]a^6 \equiv 1 \pmod 7 \Rightarrow a^6 = 7k + 1, k \in \mathbb{Z}[/itex].

Case 2: [itex]7 | a[/itex]

Hence, [itex]a^6 \equiv 0 \pmod 7 \Rightarrow a^6 = 7k, k \in \mathbb{Z}[/itex].

So [itex]a^6[/itex] is either of the form [itex]7k[/itex] or [itex]7k + 1[/itex] (QED).

You can construct something exactly like this for the squares modulo 3, and indeed generalise it for any (p-1)th power modulo a prime p.

Do you think the analogous statement for cubes would be true, given that 4 is composite?
 
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  • #3
I don't know Fermat`s little Theory, i just started number theory to be honest ..is there a possibility using div alg pls?
 
  • #4
MAGICMATHS said:
I don't know Fermat`s little Theory, i just started number theory to be honest ..is there a possibility using div alg pls?

Well, you could use the longer method presented here: http://www.mathyards.com/vb/showthread.php?691-Division-Algorithm-help !
 
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  • #5
MagicMaths, if this is homework or coursework, could you please post it in the appropriate homework forum (precalculus mathematics) in future, please?
 

Related to Solving Integer Cubes & Squares: 7k & 3k?

1. What are integer cubes and squares?

Integer cubes and squares refer to numbers that are the result of multiplying a number by itself a certain number of times. For example, 5 cubed (5^3) is equal to 5 x 5 x 5 = 125. 7 squared (7^2) is equal to 7 x 7 = 49.

2. How do you solve integer cubes and squares?

To solve integer cubes and squares, you can use the exponent rule which states that for a number raised to a power, you can multiply the base number by itself the number of times indicated by the power. For example, to solve 3 cubed (3^3), you would multiply 3 x 3 x 3 = 27. To solve 7 squared (7^2), you would multiply 7 x 7 = 49.

3. What is the difference between 7k and 3k when solving integer cubes and squares?

The difference between 7k and 3k when solving integer cubes and squares is the base number that is being multiplied. In 7k, the base number is 7, while in 3k, the base number is 3. This means that the resulting numbers will be different, but the process of solving them is the same.

4. Can you explain the significance of the numbers 7k and 3k in solving integer cubes and squares?

The numbers 7k and 3k are typically used in mathematical equations to represent unknown values or variables. In the case of solving integer cubes and squares, 7k and 3k can be used to represent any number in the form of 7^k or 3^k. This allows for a more general solution to these types of problems.

5. Are there any real-world applications of solving integer cubes and squares?

Yes, there are many real-world applications of solving integer cubes and squares. For example, in engineering and physics, these concepts are used in calculating the volume of objects or the surface area of geometric shapes. In computer science, they are used in encryption algorithms to protect sensitive information. Additionally, understanding integer cubes and squares is important in understanding more complex mathematical concepts and solving advanced equations.

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