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12. 13.
14. 15. 16. 17. Discrete Mathematics Y. Sagher . Prove that a number is rational itf its decimal representation becomes periodic. (Repeated 0’s is periodic!) . Write the number 035— as a ratio of two integers Write the number 352—7 as the ratio of two integers.
Prove that between any two rational numbers there is a rational number. Prove that between any two rational numbers there is an irrational Hum“
ber. . Let x1 = x/i and mm 2 V1 + mn_1. Prove that for all n, m” is an irrational
number.
Prove that n
. 3 ‘ 2 '
9=1
. Prove that two integers have the same remainder when divided by a third number, q, if and only if their difference is divisible by q. . Prove that a number written in the decimal system has the same remainder when divided by 9 as the sum of its digits. Prove that the sum of a rational number and an irrational number is an
irrational number. Prove that the sum of two rational numbers is a rational number.
Prove that between any two numbers there is a rational number.
Prove that between any two numbers there is an irrational number. Show that the sum of two irrational numbers can be a rational number or
an irrational number. What are all the possible remainders when you divide the square of an
integer by 5? Prove that the square root of an integer is either an integer or an irrational
number. Let d be the greatest common divisor of 102876 and 894. (a) Find d
(b) Find integers so and y so that d = 102876  2: + 894  y ...
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 Spring '08
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