Question

1. Find the integer a such that
   1. a≡-43mod23and-22≤a≤0.
   2. a≡17mod29 and-14≤a≤14.
   3. a≡-11mod21 and 90≤a≤110.

Answer 1

Answers:

a) -3

b) -12

c) 94

Explanation:-

a). a≡-43mod23 and -22≤a≤0

We can find a such that -22<=a<=22 by consecutively subtracting 23 from 43 untill we obtain value between -22 and 0.

  a≡43(mod 23)

a≡43-23(mod 23)

a≡20(mod 23)

a≡20-23(mod 23)

  a≡-3(mod 23)

As -3 between -22 and 0.

a= -3.

b).   a≡17mod29 and-14≤a≤14

We can find a such that -14≤a≤14 by consecutively subtracting 29 from 17 until we obtain a value between -14 and 14.

a≡17mod29

a≡17-29(mod29)

a≡-12mod29

As -12 between -14 and 14.

a= -12

c). a≡-11mod21 and 90≤a≤110

We can find a such that 90≤a≤110 by consecutively adding 21 to -11 untill we obtain a value between 90 and 110.

   a≡-11mod21

   a≡-11+21(mod21)

a≡10mod21

   a≡10+21(mod21)

   a≡31mod21

a≡31+21(mod21)

   a≡52mod21

   a≡52+21(mod21)

   a≡73mod21

   a≡73+21(mod21)

a≡94mod21

Since 94 between 90 and 110.

a=94