Verify that 12 2 0 (mod 3) Why is this NOT a counterexample to Fermat\'s Theorem
ID: 3031134 • Letter: V
Question
Verify that 122 0 (mod 3)
Why is this NOT a counterexample to Fermat's Theorem, which says ap1 1 (mod p)?
Because 12 is not prime. In Fermat's theorem, a must be prime.
Because 12 is divisible by 3. Fermat's theorem only applies when a is not divisible by p.
Because 12 is divisible by 2. Fermat's theorem only applies when the exponent is relatively prime to a.
It IS a counterexample. Fermat's theorem is wrong!
a)Because 12 is not prime. In Fermat's theorem, a must be prime.
b)Because 12 is divisible by 3. Fermat's theorem only applies when a is not divisible by p.
c)Because 12 is divisible by 2. Fermat's theorem only applies when the exponent is relatively prime to a.
d)It IS a counterexample. Fermat's theorem is wrong!
Explanation / Answer
If two numbers p and q are such that p – q is integrally divisible by a number r (i.e., (p-q)/r is an integer), then p and q are said to be congruent modulo r i.e. pq(mod r). As per Fermat's little theorem, if m is a prime number, then for any integer a which is not divisible by m, the number am-1-1 is an integer multiple of m, or am-1 1 mod m. Here, we are to verify that 122 0 (mod 3). Since 122 = 144 and (144- 0)/3 = 144/3 = 48, an integer, hence 122 0 (mod 3). Also 123-1 is not conruent to 1 mod 3 as 122 -1 = 143 is not divisible by 3.
This is not a counterexample to Fermat’s Theorem, as 12 is divisible by 3. The option b) is the correct answer.
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