Respuesta :
Answer:
For every n E N, when we divide n² by 7, the remainder is either 0, 1, 2, or 4.
Step-by-step explanation:
Given information: n∈N and [tex]\frac{n^2}{7}[/tex].
To prove: For every n E N, when we divide n2 by 7, the remainder is either 0, 1, 2, or 4.
Proof:
Using basic remainder remainder theorem,
[tex]remainder(\frac{n^2}{7})=(remainder(\frac{n}{7})\times remainder(\frac{n}{7}))(\text{mod }7)[/tex]
where, mod 7 is modulo 7. It means the remainder after dividing by 7.
If a natural number divide by 7 then the possible remainders are 0,1,2,3,4,5 and 6.
If remainder of n/7 is 0, then
[tex]remainder(\frac{n^2}{7})=(0\times 0)(\text{mod }7)=0[/tex]
If remainder of n/7 is 1, then
[tex]remainder(\frac{n^2}{7})=(1\times 1)(\text{mod }7)=1[/tex]
If remainder of n/7 is 2, then
[tex]remainder(\frac{n^2}{7})=(2\times 2)(\text{mod }7)=4[/tex]
If remainder of n/7 is 3, then
[tex]remainder(\frac{n^2}{7})=(3\times 3)(\text{mod }7)=9(\text{mod }7)=2[/tex]
If remainder of n/7 is 4, then
[tex]remainder(\frac{n^2}{7})=(4\times 4)(\text{mod }7)=16(\text{mod }7)=2[/tex]
If remainder of n/7 is 5, then
[tex]remainder(\frac{n^2}{7})=(5\times 5)(\text{mod }7)=25(\text{mod }7)=4[/tex]
If remainder of n/7 is 6, then
[tex]remainder(\frac{n^2}{7})=(6\times 6)(\text{mod }7)=36(\text{mod }7)=1[/tex]
For every n E N, when we divide n2 by 7, the remainder is either 0, 1, 2, or 4.
Hence proved.
