1. Modular multiplicative inverse for `11` mod `12`

Solution:
`y-=z^(-1)\ ("mod "n)-=11^(-1)\ ("mod "12)-=11\ ("mod "12)`

Extended Euclidean Algorithm

i	Quotient `q=r_1-:r_2`	Remainder `r=r_1-q*r_2`	`s=s_1-q*s_2`	`t=t_1-q*t_2`
1		`r_1=12`	`s_1=1`	`t_1=0`
2		`r_2=11`	`s_2=0`	`t_2=1`
3	`12-:11=1`	`12-1xx11=1`	`1-1xx0=1`	`0-1xx1=-1`
4	`11-:1=11`	`11-11xx1=0`	`0-11xx1=-11`	`1-11xx-1=12`

we get answer by taking the last non-zero row for Remainder `r=1` (gcd), `s=1,t=-1`

Here `t` is negative, so add 12.

`:.t=-1+12=11`

`:.` multiplicative inverse `11` mod `12=11`

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2. Modular multiplicative inverse for `7` mod `11`

Solution:
`y-=z^(-1)\ ("mod "n)-=7^(-1)\ ("mod "11)-=8\ ("mod "11)`

Extended Euclidean Algorithm

i	Quotient `q=r_1-:r_2`	Remainder `r=r_1-q*r_2`	`s=s_1-q*s_2`	`t=t_1-q*t_2`
1		`r_1=11`	`s_1=1`	`t_1=0`
2		`r_2=7`	`s_2=0`	`t_2=1`
3	`11-:7=1`	`11-1xx7=4`	`1-1xx0=1`	`0-1xx1=-1`
4	`7-:4=1`	`7-1xx4=3`	`0-1xx1=-1`	`1-1xx-1=2`
5	`4-:3=1`	`4-1xx3=1`	`1-1xx-1=2`	`-1-1xx2=-3`
6	`3-:1=3`	`3-3xx1=0`	`-1-3xx2=-7`	`2-3xx-3=11`

we get answer by taking the last non-zero row for Remainder `r=1` (gcd), `s=2,t=-3`

Here `t` is negative, so add 11.

`:.t=-3+11=8`

`:.` multiplicative inverse `7` mod `11=8`

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3. Modular multiplicative inverse for `3` mod `7`

Solution:
`y-=z^(-1)\ ("mod "n)-=3^(-1)\ ("mod "7)-=5\ ("mod "7)`

Extended Euclidean Algorithm

i	Quotient `q=r_1-:r_2`	Remainder `r=r_1-q*r_2`	`s=s_1-q*s_2`	`t=t_1-q*t_2`
1		`r_1=7`	`s_1=1`	`t_1=0`
2		`r_2=3`	`s_2=0`	`t_2=1`
3	`7-:3=2`	`7-2xx3=1`	`1-2xx0=1`	`0-2xx1=-2`
4	`3-:1=3`	`3-3xx1=0`	`0-3xx1=-3`	`1-3xx-2=7`

we get answer by taking the last non-zero row for Remainder `r=1` (gcd), `s=1,t=-2`

Here `t` is negative, so add 7.

`:.t=-2+7=5`

`:.` multiplicative inverse `3` mod `7=5`

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4. Modular multiplicative inverse for `60` mod `36`

Solution:
`y-=z^(-1)\ ("mod "n)-=60^(-1)\ ("mod "36)-=24^(-1)\ ("mod "36)-=0\ ("mod "36)`

Extended Euclidean Algorithm

i	Quotient `q=r_1-:r_2`	Remainder `r=r_1-q*r_2`	`s=s_1-q*s_2`	`t=t_1-q*t_2`
1		`r_1=36`	`s_1=1`	`t_1=0`
2		`r_2=60`	`s_2=0`	`t_2=1`
3	`36-:60=0`	`36-0xx60=36`	`1-0xx0=1`	`0-0xx1=0`
4	`60-:36=1`	`60-1xx36=24`	`0-1xx1=-1`	`1-1xx0=1`
5	`36-:24=1`	`36-1xx24=12`	`1-1xx-1=2`	`0-1xx1=-1`
6	`24-:12=2`	`24-2xx12=0`	`-1-2xx2=-5`	`1-2xx-1=3`

we get answer by taking the last non-zero row for Remainder `r=12` (gcd), `s=2,t=-1`

Here gcd (Remainder r=12) is not 1, So 60 does not have a multiplicative inverse modulo 36

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5. Modular multiplicative inverse for `35` mod `20`

Solution:
`y-=z^(-1)\ ("mod "n)-=35^(-1)\ ("mod "20)-=15^(-1)\ ("mod "20)-=0\ ("mod "20)`

Extended Euclidean Algorithm

i	Quotient `q=r_1-:r_2`	Remainder `r=r_1-q*r_2`	`s=s_1-q*s_2`	`t=t_1-q*t_2`
1		`r_1=20`	`s_1=1`	`t_1=0`
2		`r_2=35`	`s_2=0`	`t_2=1`
3	`20-:35=0`	`20-0xx35=20`	`1-0xx0=1`	`0-0xx1=0`
4	`35-:20=1`	`35-1xx20=15`	`0-1xx1=-1`	`1-1xx0=1`
5	`20-:15=1`	`20-1xx15=5`	`1-1xx-1=2`	`0-1xx1=-1`
6	`15-:5=3`	`15-3xx5=0`	`-1-3xx2=-7`	`1-3xx-1=4`

we get answer by taking the last non-zero row for Remainder `r=5` (gcd), `s=2,t=-1`

Here gcd (Remainder r=5) is not 1, So 35 does not have a multiplicative inverse modulo 20

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