Question

Consider the system of congruences in $\mathbb{Z}$ $x \equiv 1 \mod 2$ $x \equiv 2 \mod 3$ $x \equiv 3 \mod 5$ Does this system have solutions in $\mathbb{Z}$? If it does, give these solutions in the form $x \equiv a \mod n$. Hint: Use the Chinese Remainder Theorem.

Name: consider the system of congruences in mathbbz x equiv 1 mod 2 x equiv 2 mod 3 x equiv 3 mod 5 does this system have solutions in mathbbz if it does give these solutions in the form x equiv a 70716
Uploaded: 2022-05-31T09:39:06-08:00
Channel: Adi S
Description: consider the system of congruences in mathbbz x equiv 1 mod 2 x equiv 2 mod 3 x equiv 3 mod 5 does this system have solutions in mathbbz if it does give these solutions in the form x equiv a 70716

          Consider the system of congruences in $\mathbb{Z}$
$x \equiv 1 \mod 2$
$x \equiv 2 \mod 3$
$x \equiv 3 \mod 5$
Does this system have solutions in $\mathbb{Z}$? If it does, give these solutions in the form $x \equiv a \mod n$.
Hint: Use the Chinese Remainder Theorem.

consider the system of congruences in mathbbz x equiv 1 mod 2 x equiv 2 mod 3 x equiv 3 mod 5 does this system have solutions in mathbbz if it does give these solutions in the form x equiv a 70716

Added by Rebecca R.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Adi S

05/31/2022

Step 1

Substituting this into (2), we get $2k + 1 \equiv 2 \pmod{3}$ $2k \equiv 1 \pmod{3}$ Multiplying by 2, we have $4k \equiv 2 \pmod{3}$, which simplifies to $k \equiv 2 \pmod{3}$. Thus, $k = 3j + 2$ for some integer $j$. Substituting this back into the expression Show more…

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Consider the system of congruences in $\mathbb{Z}$ $x \equiv 1 \mod 2$ $x \equiv 2 \mod 3$ $x \equiv 3 \mod 5$ Does this system have solutions in $\mathbb{Z}$? If it does, give these solutions in the form $x \equiv a \mod n$. Hint: Use the Chinese Remainder Theorem.