MATH 135 Algebra, Solutions to Assignment 7
1:
(a) Find the smallest nonnegative 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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(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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 Fall '08
 ANDREWCHILDS
 Algebra, Remainder, ISBN, Prime number

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