soln7 - MATH 135 Algebra Solutions to Assignment 7 1(a Find...

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MATH 135 Algebra, Solutions to Assignment 7 1: (a) Find the smallest non-negative integer x such that x 41 (mod 9). Solution: The smallest such x is the remainder when 41 is divided by 9. We have 41 = 9 · 4 + 5, so x = 5. (b) Find the integer x which has the smallest absolute value such that x 568 (mod 41). Solution: We have 568 = 41 · 13 + 35 and so 568 35 (mod 41). Thus we have x = 568 (mod 41) ⇐⇒ x 35 (mod 41) ⇐⇒ x ∈ {··· , - 47 , - 6 , 35 , 76 , ···} . Thus the integer x with the smallest absolute value such that x 568 (mod 41) is x = - 6. (c) What day of the week will it be 1000 days after a Monday? Solution: We have 1000 = 7 · 142 + 6 so 1000 6 (mod 7). Thus 1000 days after a Monday, it will be the same day of the week as it is 6 days after a Monday, that is, it will be Sunday. (d) What time of day will it be 1000 hours after 5:00 pm? Solution: We have 1000 = 24 · 41 + 16 so 1000 16 (mod 24). Thus 1000 hours after 5:00 pm it will be the same time of day as it is 16 hours after 5:00 pm, that is, it will be 9:00 am. (e) Exactly what time of day will it be 1 million seconds after 5:00 pm? Solution: We have 1 , 000 , 000 = 60 · 16 , 666 + 40 so 1 million seconds is equal to 16,666 minutes and 40 seconds. Also, we have 16 , 666 = 60 · 277 + 46, so 16,666 minutes is equal to 277 hours and 46 minutes. Finally, we have 277 = 24 · 11 + 13 so 277 hours is equal to 11 days and 13 hours. Thus 1 million seconds is equal to 11 days, 13 hours, 46 minutes and 40 seconds. It follows that 1 million seconds after 5:00 pm, it will be the same time of day as it is 13 hours, 46 minutes and 40 seconds after 5:00 pm, that is it will be exactly 40 seconds past 6:46 am.
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2: (a) Find all positive integers m such that 126 35 (mod m ). Solution: We have 126 35 (mod m ) ⇐⇒ m ± ± (126 - 35) ⇐⇒ m ± ± 91 ⇐⇒ m ± ± 7 · 13 ⇐⇒ m = 1 , 7 , 13 or 91 . (b) Find the remainder when the integer 100 X k =1 k ! is divided by 13. Solution: We have 1! = 1
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soln7 - MATH 135 Algebra Solutions to Assignment 7 1(a Find...

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