Question

Prove that for $n \in \mathbb{Z}$, its cube $n^3$ satisfies exactly one of the following: (i) $n^3 \equiv 0 \pmod{7}$; (ii) $n^3 \equiv 1 \pmod{7}$; (iii) $n^3 \equiv 6 \pmod{7}$. Use this information to prove that 7142128354249566373 cannot be a cube.

Name: prove that for n in mathbbz its cube n3 satisfies exactly one of the following i n3 equiv 0 pmod7 ii n3 equiv 1 pmod7 iii n3 equiv 6 pmod7 use this information to prove that 7142128354249566 20377
Uploaded: 2023-04-28T03:44:53-08:00
Duration: 3 min 6 s
Channel: Madhur L
Description: prove that for n in mathbbz its cube n3 satisfies exactly one of the following i n3 equiv 0 pmod7 ii n3 equiv 1 pmod7 iii n3 equiv 6 pmod7 use this information to prove that 7142128354249566 20377

          Prove that for $n \in \mathbb{Z}$, its cube $n^3$ satisfies exactly one of the following:
(i) $n^3 \equiv 0 \pmod{7}$;
(ii) $n^3 \equiv 1 \pmod{7}$;
(iii) $n^3 \equiv 6 \pmod{7}$.

Use this information to prove that 7142128354249566373 cannot be a cube.

prove that for n in mathbbz its cube n3 satisfies exactly one of the following i n3 equiv 0 pmod7 ii n3 equiv 1 pmod7 iii n3 equiv 6 pmod7 use this information to prove that 7142128354249566 20377

Added by Donald M.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

04/28/2023

Step 1

We can check the cubes of the integers from 0 to 6 modulo 7: $0^3 \equiv 0 \pmod{7}$ $1^3 \equiv 1 \pmod{7}$ $2^3 = 8 \equiv 1 \pmod{7}$ $3^3 = 27 \equiv 6 \pmod{7}$ $4^3 = 64 \equiv 1 \pmod{7}$ $5^3 = 125 \equiv 6 \pmod{7}$ $6^3 = 216 \equiv 6 \pmod{7}$ Thus, for Show more…

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Prove that for $n \in \mathbb{Z}$, its cube $n^3$ satisfies exactly one of the following: (i) $n^3 \equiv 0 \pmod{7}$; (ii) $n^3 \equiv 1 \pmod{7}$; (iii) $n^3 \equiv 6 \pmod{7}$. Use this information to prove that 7142128354249566373 cannot be a cube.