Determine the remainder when $7^7^2019$ is divided by 47.Find the remainder when $10^400$ is divided by 199?what is the remainder when $1!+2!+3!+4!+cdots+45!$ is divided by 47?Hint for finding the remainder when $2018^2019$ is divided by 13Is there a quick way to find the remainder when this determinant is divided by $5$?The remainder when this determinant is divided by 5Determine all natural numbers which when divided by 17 leave the remainder 3 and when divided by 6 leave the remainder 1Prove that $a^2$ when divided by $5$ cannot have a remainder of $3$.Can you determine the remainder when divided by 6?Modular arithmetic, remainder of a term in a sequence when divided by 100Find the remainder when $99!+99$ is divided by $100$
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Determine the remainder when $7^7^2019$ is divided by 47.
Find the remainder when $10^400$ is divided by 199?what is the remainder when $1!+2!+3!+4!+cdots+45!$ is divided by 47?Hint for finding the remainder when $2018^2019$ is divided by 13Is there a quick way to find the remainder when this determinant is divided by $5$?The remainder when this determinant is divided by 5Determine all natural numbers which when divided by 17 leave the remainder 3 and when divided by 6 leave the remainder 1Prove that $a^2$ when divided by $5$ cannot have a remainder of $3$.Can you determine the remainder when divided by 6?Modular arithmetic, remainder of a term in a sequence when divided by 100Find the remainder when $99!+99$ is divided by $100$
$begingroup$
Determine the remainder when $7^7^2019$ is divided by 47.
47 is prime, perhaps we can do something with that? I'm not sure how to approach this question, any and all help is appreciated.
Thanks!
modular-arithmetic
asked Mar 16 at 23:50
SaniaSania
344
$endgroup$
add a comment |
$begingroup$
Determine the remainder when $7^7^2019$ is divided by 47.
47 is prime, perhaps we can do something with that? I'm not sure how to approach this question, any and all help is appreciated.
Thanks!
modular-arithmetic
asked Mar 16 at 23:50
SaniaSania
344
$endgroup$
1
$begingroup$
Think Fermat’s Little Theorem; then think about Euler’s Theorem.
$endgroup$
– Arturo Magidin
Mar 16 at 23:54
add a comment |
$begingroup$
Determine the remainder when $7^7^2019$ is divided by 47.
47 is prime, perhaps we can do something with that? I'm not sure how to approach this question, any and all help is appreciated.
Thanks!
modular-arithmetic
asked Mar 16 at 23:50
SaniaSania
344
$endgroup$
Determine the remainder when $7^7^2019$ is divided by 47.
47 is prime, perhaps we can do something with that? I'm not sure how to approach this question, any and all help is appreciated.
Thanks!
modular-arithmetic
modular-arithmetic
asked Mar 16 at 23:50
SaniaSania
344
asked Mar 16 at 23:50
SaniaSania
344
asked Mar 16 at 23:50
SaniaSania
344
asked Mar 16 at 23:50
SaniaSania
344
asked Mar 16 at 23:50
SaniaSania
344
344
1
$begingroup$
Think Fermat’s Little Theorem; then think about Euler’s Theorem.
$endgroup$
– Arturo Magidin
Mar 16 at 23:54
add a comment |
1
$begingroup$
Think Fermat’s Little Theorem; then think about Euler’s Theorem.
$endgroup$
– Arturo Magidin
Mar 16 at 23:54
1
1
$begingroup$
Think Fermat’s Little Theorem; then think about Euler’s Theorem.
$endgroup$
– Arturo Magidin
Mar 16 at 23:54
$begingroup$
Think Fermat’s Little Theorem; then think about Euler’s Theorem.
$endgroup$
– Arturo Magidin
Mar 16 at 23:54
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
By Fermat's little theorem
$$7^7^2019equiv 7^(7^2019 mod46) mod47$$
To calculate
$$7^2019 mod46$$
We have
$$7^2019equiv1^2019equiv1 mod2$$
$$7^2019equiv7^(2019 mod22)equiv7^17equiv (7)(49)^8equiv(7)(3)^8equiv(7)(27)^2(9)equiv(7)(9)(4)^2equiv19mod23$$
$$therefore 7^2019 equiv 19 mod46$$
So,
$$7^7^2019equiv 7^19equiv(7)(49)^9equiv(7)(2)^9equiv(7)(512)equiv(7)(-5)equiv-35equiv12 mod47$$
answered Mar 17 at 0:04
Peter ForemanPeter Foreman
4,1771216
$endgroup$
add a comment |
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$begingroup$
By Fermat's little theorem
$$7^7^2019equiv 7^(7^2019 mod46) mod47$$
To calculate
$$7^2019 mod46$$
We have
$$7^2019equiv1^2019equiv1 mod2$$
$$7^2019equiv7^(2019 mod22)equiv7^17equiv (7)(49)^8equiv(7)(3)^8equiv(7)(27)^2(9)equiv(7)(9)(4)^2equiv19mod23$$
$$therefore 7^2019 equiv 19 mod46$$
So,
$$7^7^2019equiv 7^19equiv(7)(49)^9equiv(7)(2)^9equiv(7)(512)equiv(7)(-5)equiv-35equiv12 mod47$$
answered Mar 17 at 0:04
Peter ForemanPeter Foreman
4,1771216
$endgroup$
add a comment |
$begingroup$
By Fermat's little theorem
$$7^7^2019equiv 7^(7^2019 mod46) mod47$$
To calculate
$$7^2019 mod46$$
We have
$$7^2019equiv1^2019equiv1 mod2$$
$$7^2019equiv7^(2019 mod22)equiv7^17equiv (7)(49)^8equiv(7)(3)^8equiv(7)(27)^2(9)equiv(7)(9)(4)^2equiv19mod23$$
$$therefore 7^2019 equiv 19 mod46$$
So,
$$7^7^2019equiv 7^19equiv(7)(49)^9equiv(7)(2)^9equiv(7)(512)equiv(7)(-5)equiv-35equiv12 mod47$$
answered Mar 17 at 0:04
Peter ForemanPeter Foreman
4,1771216
$endgroup$
add a comment |
$begingroup$
By Fermat's little theorem
$$7^7^2019equiv 7^(7^2019 mod46) mod47$$
To calculate
$$7^2019 mod46$$
We have
$$7^2019equiv1^2019equiv1 mod2$$
$$7^2019equiv7^(2019 mod22)equiv7^17equiv (7)(49)^8equiv(7)(3)^8equiv(7)(27)^2(9)equiv(7)(9)(4)^2equiv19mod23$$
$$therefore 7^2019 equiv 19 mod46$$
So,
$$7^7^2019equiv 7^19equiv(7)(49)^9equiv(7)(2)^9equiv(7)(512)equiv(7)(-5)equiv-35equiv12 mod47$$
answered Mar 17 at 0:04
Peter ForemanPeter Foreman
4,1771216
$endgroup$
By Fermat's little theorem
$$7^7^2019equiv 7^(7^2019 mod46) mod47$$
To calculate
$$7^2019 mod46$$
We have
$$7^2019equiv1^2019equiv1 mod2$$
$$7^2019equiv7^(2019 mod22)equiv7^17equiv (7)(49)^8equiv(7)(3)^8equiv(7)(27)^2(9)equiv(7)(9)(4)^2equiv19mod23$$
$$therefore 7^2019 equiv 19 mod46$$
So,
$$7^7^2019equiv 7^19equiv(7)(49)^9equiv(7)(2)^9equiv(7)(512)equiv(7)(-5)equiv-35equiv12 mod47$$
answered Mar 17 at 0:04
Peter ForemanPeter Foreman
4,1771216
answered Mar 17 at 0:04
Peter ForemanPeter Foreman
4,1771216
answered Mar 17 at 0:04
Peter ForemanPeter Foreman
4,1771216
answered Mar 17 at 0:04
Peter ForemanPeter Foreman
4,1771216
4,1771216
add a comment |
add a comment |
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$begingroup$
Think Fermat’s Little Theorem; then think about Euler’s Theorem.
$endgroup$
– Arturo Magidin
Mar 16 at 23:54