Prove that $a + 2b equiv 0 pmod{3}$ if and only if $a equiv bpmod{3}$.
$begingroup$
I need to prove that $a + 2b equiv 0pmod{3}$ if and only if $a equiv b pmod{3}$.
I know that you need to show both cases but my professor said that we weren't supposed to use one to solve the other so I'm stuck.
elementary-number-theory
edited Dec 9 '18 at 2:38
Shaun
9,000113682
asked Dec 9 '18 at 1:53
Lauren YLauren Y
61
$endgroup$
add a comment |
$begingroup$
I need to prove that $a + 2b equiv 0pmod{3}$ if and only if $a equiv b pmod{3}$.
I know that you need to show both cases but my professor said that we weren't supposed to use one to solve the other so I'm stuck.
elementary-number-theory
edited Dec 9 '18 at 2:38
Shaun
9,000113682
asked Dec 9 '18 at 1:53
Lauren YLauren Y
61
$endgroup$
add a comment |
$begingroup$
I need to prove that $a + 2b equiv 0pmod{3}$ if and only if $a equiv b pmod{3}$.
I know that you need to show both cases but my professor said that we weren't supposed to use one to solve the other so I'm stuck.
elementary-number-theory
edited Dec 9 '18 at 2:38
Shaun
9,000113682
asked Dec 9 '18 at 1:53
Lauren YLauren Y
61
$endgroup$
I need to prove that $a + 2b equiv 0pmod{3}$ if and only if $a equiv b pmod{3}$.
I know that you need to show both cases but my professor said that we weren't supposed to use one to solve the other so I'm stuck.
elementary-number-theory
elementary-number-theory
edited Dec 9 '18 at 2:38
Shaun
9,000113682
asked Dec 9 '18 at 1:53
Lauren YLauren Y
61
edited Dec 9 '18 at 2:38
Shaun
9,000113682
asked Dec 9 '18 at 1:53
Lauren YLauren Y
61
edited Dec 9 '18 at 2:38
Shaun
9,000113682
edited Dec 9 '18 at 2:38
Shaun
9,000113682
edited Dec 9 '18 at 2:38
Shaun
9,000113682
9,000113682
asked Dec 9 '18 at 1:53
Lauren YLauren Y
61
asked Dec 9 '18 at 1:53
Lauren YLauren Y
61
asked Dec 9 '18 at 1:53
Lauren YLauren Y
61
61
add a comment |
add a comment |
4 Answers
4
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oldest
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$begingroup$
Hint: $ 2equiv -1pmod3 $ so $ a+2bequiv a-bpmod 3 $
edited Dec 9 '18 at 2:39
Bill Dubuque
210k29192640
answered Dec 9 '18 at 1:55
Tsemo AristideTsemo Aristide
57.5k11444
$endgroup$
add a comment |
$begingroup$
Hint: Try converting the second equation to
$$
a -b equiv 0 pmod 3
$$
by subtracting $b$ from both sides.
edited Dec 9 '18 at 2:38
Bill Dubuque
210k29192640
answered Dec 9 '18 at 1:55
John HughesJohn Hughes
63.3k24090
$endgroup$
add a comment |
$begingroup$
$a equiv b pmod 3 iff 3|a-b iff 3|a-b + 3b iff 3|a+2b iff a+2b equiv 0 pmod 3$
or...
$2equiv -1 pmod 3$ so....
$a+2b equiv 0pmod 3 iff a- b equiv 0 pmod 3 iff a equiv b pmod 3$
answered Dec 9 '18 at 1:57
fleabloodfleablood
69.6k22685
$endgroup$
add a comment |
$begingroup$
Hint: Here $2$ is "a multiple of three away from" $-1$, meaning that they are what modulo three?
They are equivalent.
answered Dec 9 '18 at 2:26
ShaunShaun
9,000113682
$endgroup$
add a comment |
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4 Answers
4
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4 Answers
4
active
oldest
votes
active
oldest
votes
active
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votes
$begingroup$
Hint: $ 2equiv -1pmod3 $ so $ a+2bequiv a-bpmod 3 $
edited Dec 9 '18 at 2:39
Bill Dubuque
210k29192640
answered Dec 9 '18 at 1:55
Tsemo AristideTsemo Aristide
57.5k11444
$endgroup$
add a comment |
$begingroup$
Hint: $ 2equiv -1pmod3 $ so $ a+2bequiv a-bpmod 3 $
edited Dec 9 '18 at 2:39
Bill Dubuque
210k29192640
answered Dec 9 '18 at 1:55
Tsemo AristideTsemo Aristide
57.5k11444
$endgroup$
add a comment |
$begingroup$
Hint: $ 2equiv -1pmod3 $ so $ a+2bequiv a-bpmod 3 $
edited Dec 9 '18 at 2:39
Bill Dubuque
210k29192640
answered Dec 9 '18 at 1:55
Tsemo AristideTsemo Aristide
57.5k11444
$endgroup$
Hint: $ 2equiv -1pmod3 $ so $ a+2bequiv a-bpmod 3 $
edited Dec 9 '18 at 2:39
Bill Dubuque
210k29192640
answered Dec 9 '18 at 1:55
Tsemo AristideTsemo Aristide
57.5k11444
edited Dec 9 '18 at 2:39
Bill Dubuque
210k29192640
edited Dec 9 '18 at 2:39
Bill Dubuque
210k29192640
edited Dec 9 '18 at 2:39
Bill Dubuque
210k29192640
210k29192640
answered Dec 9 '18 at 1:55
Tsemo AristideTsemo Aristide
57.5k11444
answered Dec 9 '18 at 1:55
Tsemo AristideTsemo Aristide
57.5k11444
answered Dec 9 '18 at 1:55
Tsemo AristideTsemo Aristide
57.5k11444
57.5k11444
add a comment |
add a comment |
$begingroup$
Hint: Try converting the second equation to
$$
a -b equiv 0 pmod 3
$$
by subtracting $b$ from both sides.
edited Dec 9 '18 at 2:38
Bill Dubuque
210k29192640
answered Dec 9 '18 at 1:55
John HughesJohn Hughes
63.3k24090
$endgroup$
add a comment |
$begingroup$
Hint: Try converting the second equation to
$$
a -b equiv 0 pmod 3
$$
by subtracting $b$ from both sides.
edited Dec 9 '18 at 2:38
Bill Dubuque
210k29192640
answered Dec 9 '18 at 1:55
John HughesJohn Hughes
63.3k24090
$endgroup$
add a comment |
$begingroup$
Hint: Try converting the second equation to
$$
a -b equiv 0 pmod 3
$$
by subtracting $b$ from both sides.
edited Dec 9 '18 at 2:38
Bill Dubuque
210k29192640
answered Dec 9 '18 at 1:55
John HughesJohn Hughes
63.3k24090
$endgroup$
Hint: Try converting the second equation to
$$
a -b equiv 0 pmod 3
$$
by subtracting $b$ from both sides.
edited Dec 9 '18 at 2:38
Bill Dubuque
210k29192640
answered Dec 9 '18 at 1:55
John HughesJohn Hughes
63.3k24090
edited Dec 9 '18 at 2:38
Bill Dubuque
210k29192640
edited Dec 9 '18 at 2:38
Bill Dubuque
210k29192640
edited Dec 9 '18 at 2:38
Bill Dubuque
210k29192640
210k29192640
answered Dec 9 '18 at 1:55
John HughesJohn Hughes
63.3k24090
answered Dec 9 '18 at 1:55
John HughesJohn Hughes
63.3k24090
answered Dec 9 '18 at 1:55
John HughesJohn Hughes
63.3k24090
63.3k24090
add a comment |
add a comment |
$begingroup$
$a equiv b pmod 3 iff 3|a-b iff 3|a-b + 3b iff 3|a+2b iff a+2b equiv 0 pmod 3$
or...
$2equiv -1 pmod 3$ so....
$a+2b equiv 0pmod 3 iff a- b equiv 0 pmod 3 iff a equiv b pmod 3$
answered Dec 9 '18 at 1:57
fleabloodfleablood
69.6k22685
$endgroup$
add a comment |
$begingroup$
$a equiv b pmod 3 iff 3|a-b iff 3|a-b + 3b iff 3|a+2b iff a+2b equiv 0 pmod 3$
or...
$2equiv -1 pmod 3$ so....
$a+2b equiv 0pmod 3 iff a- b equiv 0 pmod 3 iff a equiv b pmod 3$
answered Dec 9 '18 at 1:57
fleabloodfleablood
69.6k22685
$endgroup$
add a comment |
$begingroup$
$a equiv b pmod 3 iff 3|a-b iff 3|a-b + 3b iff 3|a+2b iff a+2b equiv 0 pmod 3$
or...
$2equiv -1 pmod 3$ so....
$a+2b equiv 0pmod 3 iff a- b equiv 0 pmod 3 iff a equiv b pmod 3$
answered Dec 9 '18 at 1:57
fleabloodfleablood
69.6k22685
$endgroup$
$a equiv b pmod 3 iff 3|a-b iff 3|a-b + 3b iff 3|a+2b iff a+2b equiv 0 pmod 3$
or...
$2equiv -1 pmod 3$ so....
$a+2b equiv 0pmod 3 iff a- b equiv 0 pmod 3 iff a equiv b pmod 3$
answered Dec 9 '18 at 1:57
fleabloodfleablood
69.6k22685
answered Dec 9 '18 at 1:57
fleabloodfleablood
69.6k22685
answered Dec 9 '18 at 1:57
fleabloodfleablood
69.6k22685
answered Dec 9 '18 at 1:57
fleabloodfleablood
69.6k22685
69.6k22685
add a comment |
add a comment |
$begingroup$
Hint: Here $2$ is "a multiple of three away from" $-1$, meaning that they are what modulo three?
They are equivalent.
answered Dec 9 '18 at 2:26
ShaunShaun
9,000113682
$endgroup$
add a comment |
$begingroup$
Hint: Here $2$ is "a multiple of three away from" $-1$, meaning that they are what modulo three?
They are equivalent.
answered Dec 9 '18 at 2:26
ShaunShaun
9,000113682
$endgroup$
add a comment |
$begingroup$
Hint: Here $2$ is "a multiple of three away from" $-1$, meaning that they are what modulo three?
They are equivalent.
answered Dec 9 '18 at 2:26
ShaunShaun
9,000113682
$endgroup$
Hint: Here $2$ is "a multiple of three away from" $-1$, meaning that they are what modulo three?
They are equivalent.
answered Dec 9 '18 at 2:26
ShaunShaun
9,000113682
answered Dec 9 '18 at 2:26
ShaunShaun
9,000113682
answered Dec 9 '18 at 2:26
ShaunShaun
9,000113682
answered Dec 9 '18 at 2:26
ShaunShaun
9,000113682
9,000113682
add a comment |
add a comment |
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