Bunuel wrote:
If \(P = k^3 – k\), where k and P are positive integers, and k is given by the expression \(k = (10x)^{n} + 5^{4}\), where x and n are positive integers, then what is the remainder when P is divided by 4 ?
A. 0
B. 1
C. 2
D. 3
E. Cannot be determined
Are You Up For the Challenge: 700 Level QuestionsQuick one.
\(P = k^3 – k\) says that P is product of three consecutive integers. and \(k = (10x)^{n} + 5^{4}\) that the unit digit of k is 5. Thus P Must be a multiple of 8 since it's last two digits is a MULTIPLE OF 8.
4 is a factor of 8 thus A is the answer.
LET'S EXPLAIN IT ANOTHER WAY
The product of three consecitive positive integers must be a multiple of at least 6 and in one situation of 8. This is because any three consecutive postive integer must contain a multiple of 3(because you're counting up to three consecutiveness) and an even number which means a multiple of 2; \((3*2 = 6)\)
The only situation where consecutive integers (k-1), k, and (k+1) must be a multiple of 8 is when k has a unit digit of 5, as in 5 or 15 or 25 etc.. thus multiplying whatever that ends in 5 with the preceding consecutive number and next consecutive must contain up to three different 2s which is 8 when multiplied.
\(k = (10x)^{n} + 5^{4}\) plain tells you the unit digit of k is 5. So P MUST be divisible by 8.