1. use the definition of odd and even integers to prove that if x is an odd inte
ID: 2902754 • Letter: 1
Question
1. use the definition of odd and even integers to prove that if x is an odd integer and y is an even integer, then 9x+5y-20 is an odd integer.
2. use the definition of rational numbers to prove that if u and v are rational then 8u+(3/2)v is rational.
3. Use the DEFINITION of divisibility to prove the following statement:
Let a; b; c = Z with a != 0. If a and ac then a(6a + 5b - 3c).
4. Use the DEFINITION of congruences to prove the following statement:
Let n = N and a; b; c; d = Z. If a =b (mod n) and c = d (mod n) then
2a - 7c = 2b - 7d (mod n).
Explanation / Answer
1.
we know that x is odd and y is even
let x = 2a+1, y = 2b for some intergers a,b
=>
9x+5y-20 = 9*(2a+1) +5*2b -20 = 18a +9 +10b -20 = 18a + 10b -11 = 2*(9a + 5b-6) +1
=>
9x+5y-20 is odd
thus proved
2.
u, v are rational
=>
there exists integers a,b,c,d such that
u = a/b, v = c/d
=>
8u +3/2v = 8a/b + 3c/2d = (16ad + 3bc)/(2bd)
we can see that 16ad+3bc, 2bd are integers
=>
8u+3v/2 is a rational number by definiton
thus proved
3.
let a|b, a|c
=>
b = p*a, c = q*a for some integers p,q
=>
6a+5b-3c = 6a+ 5pa -3qa = a*(6+5p-3q)
=>
a| (6a+5b-3c) since 6+5p-3q is an integer
thus proved
4.
a= b (mod n),
c= d(mod n)
=>
n|(a-b), n|(c-d)
=>
n|2*(a-b), n|7(c-d)
=>
n|[2*(a-b)-7(c-d)]
=>
n|[(2a-7c) - (2b-7d)]
=>
2a-7c = 2b-7d (mod n)
thus proved
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