1 Let a and b be integers prove that if a b then a bc for a

1) Let a and b be integers, prove that if a| b then a | (b*c) for any integer c.

2) Find the quotients and remainders

a) 101 div 7

b) 101 mod 7

c) -101 div 11

d) -101 mod 11

3) Today is a Thursday, what is the day in week (i.e., Monday, Tuesday, …) after 200 days from now? Use congruence to explain you answer.

Solution

Let, a|b Then b = ka for some integer k

Multiplying c in both side we get...

b*c = ka*c
or, bc= kc*a
Hence a|bc

2) Find the quotients and remainders
a) 101 div 7
quotient = 14

b) 101 mod 7
remainder = 3

c) -101 div 11
quotient = -9

d) -101 mod 11
remainder = -2

3) Today is a Thursday, what is the day in week after 200 days from now?

Today is Thursday
1 day from today\'s date is Friday
2 days from today\'s date is Satur.day
3 days from today\'s date is Sunday
4 days from today\'s date is Monday
5 days from today\'s date is Tuesday
6 days from today\'s date is Wednesday
7 days from today\'s date is Thursday
14 = 7*2 days from today\'s date is Thursday
21 = 7*3 days from today\'s date is Thursday
...
70 = 7*10 days from today\'s date is Thursday
...
140 = 7*20 days from today\'s date is Thursday
196 = 7*28 days from today\'s date is Thursday
197 days from today\'s date is Friday
198 days from today\'s date is Satur.day
199 days from today\'s date is Sunday
200 days from today\'s date is Monday

So, 200 days from today\'s date is Monday