week 2: number theory practice questions pending

remainders/residues/residue classes

Q1. If a = b mod m and c = d mod m, then:
Prove that:
(1) a +- c = b +- d mod m
(2) ac = bd mod m
(3) ax + cy = bx + dy mod m

Q2. if a = b mod m, pr. th. a^n = b^n mod m.

Q3. If a = b mod m then a = b mod d if d | m.

Q4. 
Remainder of 13^73 + 14^3 mod 11.

Q5.
Pr. th.
ax = ay mod m iff x = y mod (m/gcd(a,m))
S5.
m = k.q1, a = k.q2 where k = gcd(a,m)
m/k = q1
x = y mod q1 => ax = ay mod q1 => ax = ay mod m since q1| m.
------------------------------
Homework multiple choice:

Q1. True/False

For a polynomial f(x) with integer coefficients and integers a,b,l 
we have a = b mod l => f(a) = f(b) mod l?
S1.
True

Q2.

The 2-digit integers from 19 to 92 are written consecutively to form the integer 

N = 192021⋯9192. Suppose that 3^k is the highest power of 3 that is a factor of N.
What is k?

S2.
Method 1:
Since 10^m = 1 mod 9 for any m
=> 192021⋯9192 = 19 + 20 .. 92 mod 9 = 74/2[19 + 92] = 37*111
mod 3 = 0 
mod 9 = 1 * 3 = 3
So k = 1.

Method 2:
Sum of digits =  (2+3+4+5+6+7+8)*10 + (1+2... 9)*7 + sum_digits(19 + 90 + 91 + 92)
= 35*10 + 45*7 + 40
Doing mod 3 gives 2 + 0 + 1 = 0 mod 3
Doing mod 9 gives 8 + 0 + 4 = 3 mod 9

So 3 divides it but 9 doesn't. k = 1 = answer.

Q3. Remainder of 3^89*7^86 mod 17?
S3.
3^4 = 81 = -4 m 17
3^8 = 16 m 17 = -1
3^89 = 3^88.3 = (-1)^11.3 mod 17 = -3
7^2 = 49 = -2 m 17
7^8 = 16 m 17 = -1
7^80 = (-1)^10
7^86 = (-2)^3 m 17 = -8
So -8*-3 m 17 = 24 = 7 m 17


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