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Solution
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Solution provided by AtoZmath.com
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Modulo calculator
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1. Chinese Remainder Theorem
 1. x=2 (mod 5),x=3 (mod 7),x=10 (mod 11)
2. x=4 (mod 10),x=6 (mod 13),x=4 (mod 7),x=2 (mod 11)
3. x=6 (mod 11),x=13 (mod 16),x=9 (mod 21),x=19 (mod 25)
4. 3x=7 (mod 10)
5. 3x=6 (mod 12)
6. 2x=5 (mod 7),3x=4 (mod 8)
7. 2x=1 (mod 3),3x=5 (mod 8)
8. 2x=6 (mod 14),3x=9 (mod 15),5x=20 (mod 60)
9. x=3 (mod 7),x=3 (mod 5),x=4 (mod 12)
10. x=1 (mod 4),x=0 (mod 6)
2. Modulo
1. 42 mod 5
2. 3^302 mod 5
3. 19^24 mod 21
4. 7^106 mod 143
5. 27^400 mod 619
6. 42^-1 mod 5
3. Extended Euclidean Algorithm, Euclid's Algorithm, Modular multiplicative inverse
1. 11 and 12
2. 7 and 11
3. 3 and 7
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Example1. 3^302 mod 5
Solution:
`3^302" mod "5`
Here `3^302=(3^2)^151`
`=(3^2" mod "5)^151" mod "5`
`=(9" mod "5)^151" mod "5`
`=4^151" mod "5`
Here `4^151=(4^2)^75*4`
`=(((4^2" mod "5)^75" mod "5)*(4" mod "5))" mod "5`
`=(((16" mod "5)^75" mod "5)*4)" mod "5`
`=((1^75" mod "5)*4)" mod "5`
`=((1" mod "5)*4)" mod "5`
`=(1*4)" mod "5`
`=4" mod "5`
`=4` |
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