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1. Show that if = 8, then both X and y are odd. 2. Find all integer solutions of the equation 4x + 51y = 9. 3. Using the Chinese Remainder Theorem, solve the system x=3(mod 5) x=2 (mod 7) x=1 (mod 4) 4. Determine the last two digits of 9764002 5. Does x²=11 (mod 127) have a solution? 6. Use the continued fraction expansion of V37 to generate a solution to the equation x²-37y²=1. 7. Find the remainder of the division of 3201 with 1901. [Notice that 1901 is prime.) 8. Determine the number of terminal zeros of 500!/200!.

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8) 500!/200!= 201*202*…*500
We notice the following pattern: the product of ten consecutive numbers starting from 201 to 210 ends in two zeros (one comes from 202*205 and the 2nd from the 10th number – 210). This pattern holds up to 291*…*300 (where we have three 0’s instead). Then, between 301*..*310 and so on (up to 400) the same number of 0’s occurs...

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