Combinatorics | Number Theory Answers

how many four digit positive integers that are divisible by 3 can be formed using the digit 0,1,2,3,4,5 and 6, without any of the digits getting repeating ?

what is hte answer and reason of 2=3

I have been trying to roll a 7 dice yahtzee on one roll, and I ended up rolling a 6 of a kind with 7 dice, a yahtzee with 5 dice is 1/1296, are the odds of what I rolled higher or lower then that?

Let (G, ∗) be the group of all number theoretic functions f : N → C satisfying f (1) ̸= 0, with group operation given by the Dirichlet product ∗. The identity function I is the identity element of G defined by I : N → Z where I={1 if n=1, 0 if n>1}

An element f ∈ G is said to have finite order if there exists an n∈N such that f^n =I (here f^n =f∗f∗···∗f with n factors). Find all number theoretic functions f : N → Z with f(1) ≠ 0 that have finite order in the group G.

Well now, I thought to start from the fact that since f(1)=/0, then f has an inverse called g. I know that f*g=I under Dirichlet product so if I need to find f such that f*f*f...*f =I means that f^(n-1) is my inverse function but that's just what I can came up with after hours , with the info I have from my lecture notes . If anyone can hep would be much appreciated.

A,B,C are in Arithmetic Progression(A.P.)
Prove that A(1/B + 1/C), B(1/C+ 1/A) , C(1/A +1/B) are also in AP.

A gardener throws 18 seeds onto an equilateral triangle shaped plot of land with sides of length one metre. Then at least two seeds are within a distance of 25 centimetres.true/false