数论or数的性质

数论or数的性质，兼答天山真题
onion 2004-01-10 11:53:24
数论 or 数的性质

如果n是一个正整数，r是(n+1)(n-1)除以24的余数，r的值是多少？
a 2不是n的一个因子
b 3不是n的一个因子

I do not think it is hard for everyone to acknowledge neither of the conditions is sufficient. So let us talk about if both of them are.

According to a and b, n could not be divided exactly by 2, 3 and 6 [so obviously, right?]. Then let us talk about the possibility of n, I would like to define n = 6k + n%6 [remainder of n divided by m]

So n only could be n = 6k + 1, or 6k + 5 [or 6k -1]

Because, n could not be as follows:
If n = 6k +2, or n =6k+4, n%2 =0;
If n = 6k +3, n%3 =0;

So let us pick n = 6k+1,
(n+1)(n-1) = (6k+2) (6k) = 12k(3k+1).
Let us talk about k,
if k is an even #, 12k%24 = 0;
if not, 12(3k+1)%24=0. [3k+1 is even, obviously, right?]
So no matter what n could be, (n+1) (n-1) %24 =0.

[the same for n = 6k-1]

Therefore, r must be 0 for (n+1)(n-1) could be divided exactly by 24.

　

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