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+8 votes
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The minimum positive integer p such that 3p modulo 17 = 1 is

  1. 5
  2. 8
  3. 12
  4. 16
asked in Set Theory & Algebra by Veteran (21.5k points)
retagged by | 876 views

3 Answers

+8 votes
Best answer

Fermat's Little Theorem :

a≡ a (mod p)                        

According to Modular Arithmetic    ≡ b (mod n) if their difference (a-b) is an integer multiple of n ( n divides (a-b) )

So ( ap - a ) is an integer multiple of  p , now as a is not divisible by p so definitely  ( ap-1 -1) is an integer multiple of p .this simply means if we divides ap-1  by p , the remainder would be 1 .... ap-1 modulo p = 1 

put the values in the formula.          p=17 so p-1 =16 .

answered by Junior (579 points)
selected by
+7 votes
D) fermat's little theorem
answered by Boss (7.1k points)

How (C)

I am getting (D)

What is the need to know any theorem !!

Can't we do directly by putting 'p' ??

 

offcourse you can!
But little ferment does not guarantee that you will get minimum p.
0 votes

Using Fermats Little Theorem

p: prime 
a : integer Not prime 
then
ap-1 mod p is always 1 

Here p : 7  Hence p-1  is 16

answered by Loyal (3.2k points)

Ur definition of a is wrong

a is any integer which is not divisible by p.

 

And 1 more typo is there p=17 not 7

Plz correct it.

 

But it doesnt guarantee that it will be minimum,or does it?


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