Show that every integer of the form 6k+5 must also be of the form 3k+2…
Setting we have immediately…
Regards
Some number theory problems:
1. Show that every integer of the form 6k+5 must also be of the form 3k+2
2. Show that the sqaure of any odd integer is of the form 8k+1
3. Prove that the cube of any integer can be written as one of 9k-1,9k or 9k+1
4. Prove that the sum of the squares of two odd integers cannot be a perfect square.
5. Prove that the difference of two consectutive cubes is odd.
Any help would be gratefully accepted.
Cheers
Cabouli
Hello, Cabouli!
Here are a few of them . . .
An odd integer has the form: .2. Show that the square of any odd integer is of the form
Its square is: .
We have: .
Note that and are consecutive integers.
So one of them is even and the other is odd.
Hence: . is an even integer,
And we have: .
If an integer is even, ,4. Prove that the sum of the squares of two odd integers cannot be a perfect square.
. . its square is: . , a multiple of 4.
If an integer is odd, ,
. . its square is: . , one more than a multiple of 4.
Hence, the square of an integer has the form
The sum of the squares of two odd integers is:
. .
This is two more than a multiple of 4 . . . It cannot be a square.
The two consecutive cubes are: .5. Prove that the difference of two consectutive cubes is odd.
Their difference is: .
The product of two consecutive integers is even,