Detemine (with proof ) whether is a subset X of the integers with the following property : for any intefer n there is exactyly one solution of a +2b = n with a,b \(\in\) X
Determine which binominal coefficients \(_{C^n_r=\dfrac{n!}{r!\left(n-r\right)!}}\)are odd.
Let p an odd prime number. For any positive integer k, show that there exists a positive integer m such that the rightmost k digits of \(m^2\), when expressed in the base p, are all 1's.
A 10-digit nukber is called a cute number if its digits belong to the set {1,2,3} and the difference of every pair consecutive digits is 1.
a, Find the total number of cute numbers.
b, Prove that the sum of all cute numbers is divisible by 1408.
