1. Prove that if a positive integer is of the form 6q+5 then it is of the form 3q+2 for some integer q but not conversely.
2. Prove that root p + root q is irrational where p and q are primes.
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Answers
Answered by
42
Let a be any positive integer and b = 6
so using euclids algorithm a = 6q + r
for some integer q ≥ 0 where r = 0,1,2,3,4,5 so r lie from 0≤r≤6
so a = 6q , a=6q+1 , a=6q+2 , a=6q +3 , a=6q+4,a=6q+5,a=6q+5
we can say
6q+1 = 2 so 3q +1 = 2k1 + 1 which is positive integer
(6q +3)=(6q+2)+1=2(3q+1)+1 = 2k2+1 where k2 is an integer
(6q+5)=(6q+4) +1 =2(3q+2)+1=2k3 +1 where k is an integer
so we can say that
6q+1 ,6q+3 ,6q+5 are integers
so
6q+1,6q+3,6q+5 are not exactly divisible by 2 u may know that these expression is used for the odd numbers or therefore any odd integer can be expressed by n+1 ,n+2 ,n+3 where n = 6q
so hence prove i hope u are agree with my ans
so using euclids algorithm a = 6q + r
for some integer q ≥ 0 where r = 0,1,2,3,4,5 so r lie from 0≤r≤6
so a = 6q , a=6q+1 , a=6q+2 , a=6q +3 , a=6q+4,a=6q+5,a=6q+5
we can say
6q+1 = 2 so 3q +1 = 2k1 + 1 which is positive integer
(6q +3)=(6q+2)+1=2(3q+1)+1 = 2k2+1 where k2 is an integer
(6q+5)=(6q+4) +1 =2(3q+2)+1=2k3 +1 where k is an integer
so we can say that
6q+1 ,6q+3 ,6q+5 are integers
so
6q+1,6q+3,6q+5 are not exactly divisible by 2 u may know that these expression is used for the odd numbers or therefore any odd integer can be expressed by n+1 ,n+2 ,n+3 where n = 6q
so hence prove i hope u are agree with my ans
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