Lol shoe very good shoe

Mathematics

2 answers:

Tanya [424]3 years ago

6 0

Shoe look good for a shoe

steposvetlana [31]3 years ago

6 0

Amazing shoe my guy, truly a masterpiece

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Let n be a natural number. Show that 3 | n3 −n

Vladimir [108]

Let $n=1$ . Then $n^3-n=1^3-1=0$ . By convention, every non-zero integer $n$ divides 0, so $3\vert n^3-n$ .

Suppose this relation holds for $n=k$ , i.e. $3\vert k^3-k$ . We then hope to show it must also hold for $n=k+1$ .

You have

$(k+1)^3-(k+1)=(k^3+3k^2+3k+1)-(k+1)=(k^3-k)+3(k^2+k)$

We assumed that $3\vert k^3-k$ , and it's clear that $3\vert 3(k^2+k)$ because $3(k^2+k)$ is a multiple of 3. This means the remainder upon divides $(k+1)^3-(k+1)$ must be 0, and therefore the relation holds for $n=k+1$ . This proves the statement.

4 0

2 years ago

Determine if the statement below is always true, sometimes true, or never true. When x is a

andrew11 [14]

Answer:

Sometimes

Step-by-step explanation:

When x is a nonzero integer, then x can be