Number Theory and Cryptography
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Number Theory and Cryptography MATH2088/2988
1. Find gcd(1020, 84).
Answer
2. Find the smallest prime which divides 123456789123456789.
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3. Find which element of {1, 2, · · · , 58} is inverse to 17 modulo 59.
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MATH2088/2988: Sample Quiz Page 2 of 2
MATH2088/2988 Sample Quiz
4. Find the order of 5 modulo 31.
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5. Find the residue of 31010 modulo 7.
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6. Find the unique x 2 {0, 1, 2, · · · , 194} such that x ⌘ 3 (mod 13) and x ⌘ 2 (mod 15).
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MATH2088/2988: Sample Quiz Page 3 of 2
Number Theory and Cryptography MATH2088/2988
7. Find the residue of 21010 modulo 111.
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8. Find (640), the sum of the positive integer divisors of 640.
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9. What is the smallest positive integer with exactly 10 positive divisors?
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MATH2088/2988: Sample Quiz Page 4 of 2
MATH2088/2988 Sample Quiz
10. If a simple substitution cipher encrypts the word SUGAR as JWZXD, what is the decryption of XDZWJ?
Answer