Just a Mod Problem...
The number of factors of 1 0 ( 4 5 1 1 − 1 ) is some number x. If
x ≡ a ( m o d 6 ) ,
then find the value of "a".
Details and assumptions:
-1 < a < 6 , for "a" an integer.
The answer is 3.
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1 solution
Or you may use the Chinese Remainder Theorem on the following system of congruencies to find 3 1 1 ≡ ? ( m o d 6 ) :
{ 3 1 1 ≡ 0 ( m o d 3 ) 3 1 1 ≡ 1 ( m o d 2 )
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great genius
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Nah. I'm pretty dumb in general.
Note that you can get the final answer even without using CRT if you use some logical reasoning. The first congruence implies that the answer is either 0 or 3 . The second congruence rules out the answer 0 since it's even and hence we have our answer as 3 .
The number of factors of:
1 0 4 5 1 1 − 1
Can be expressed as:
5 4 5 1 1 − 1 ⋅ 2 4 5 1 1 − 1
Using the property that the number of factors of a number is the product of the powers of its prime factorization with 1 added to each power, the number of factors turns out to be:
( 4 5 1 1 ) 2
From that, the modulus can be readily solved.
( 4 5 1 1 ) 2 = 4 5 2 2
4 5 2 2 ≡ 3 2 2 ( m o d 6 )
Now, it can be seen that no matter the power that three is raised to, with a modulus of 6, the residue/remainder will always be three. That can be seen by the fact that 3 mod 6 is equivalent to three, as is 9 mod 6, as is 27 mod 6... If you are familiar with cycling, then the idea is used here, with a cycle length of one. Since the cycle length is one, any power of 3 is equivalent to 3 when a modulus of 6 is applied.
Thus, the answer is 3.