Questions of the week are updated every sunday so make sure you check in.
The solutions will also be posted
anyone who solves this question before sunday, email me and i'll post your name.
the starred number indicates it's level of hardness (scale 1-5)

Week Three
(*2)
Express 0.00388888888888888(infinitly many 8's)as a fraction in lowest terms, and calculate the sum of the numrator and denominator.
solution on sunday


Week Two
(*2)
What is the largest integer x for whick 1/x is larger than 4/49?
solution 1/x > 4/49, so x < 49/4 (we reverse the sign(>) when taking the reciprocal)
(49/4= 12.25)
The largest such x is 12


Week Two
(*4)
prove that n5 - n is divisible by 10 for all integers n.
solution lets see the factorization.
  n5 - n = n(n4-1) = n(n2+1)(n2-1)= Tn(n2+1)(n+1)(n-1).
To show that the number is divisible by 10, we must show that it's divisible by 2 and 5. If n is even then the whole number also even and divisible by two.
If n is odd then (n+1) is even so the whole number is also even.
to prove that n5 -n is divisible by 5 we need to check if n = 0 mod 5.
n=0 mod 5 = 0-0 = 0 mod 5
n=1 mod 5 = 1-1 = 0 mod 5
n=2 mod 5 = 32-2 = 30 =0 mod 5
n=3 mod 5 = 243 - 3 = 240 =0 mod 5
n=4 mod 5 = 1024 - 4 = 1020= 0 mod 5

we have checked all the possibilities,proving that the number is divisible by 5.
we proved that the number is divisible by 2 and 5, thus it's divisible by 10.
If you do not quite understand this Solution, you should visit my mods tutorial



home