A US Social Security number is a nine-digit number. The first digit (or digits) may be zero.
a) How many US Social Security numbers are available?
b) How many US Social Security numbers are odd?
c) How many US social security numbers read the same backward and forward (eg 350767053)?

These 4-digit numbers cannot start with zero.
a) How many 4-digit numbers do not contain number 5 among its digits?
b) How many 4-digit numbers contain at least one number 5 among its digits?
c) How many 4-digit numbers contain exactly one number 5 among its digits?

Prove that if a is an odd integer, then a^2 + 7 is divisible by 4.

Read very carefully each of these sentences. Then, complete them:
a) The only integer which is divisible by every other integer is...
b) The integers that divide all prime numbers are ... and ...
c) If p is a prime number, it has ... positive divisors, and ... integer divisors.
d) (See the attachment for the equation)
e) Which one is larger, 2^1000 or 500! ? The answer is ...
f) The set {{2,6},{3,6}} has exactly ... elements. Does it coincide with {2,3,6}?

For each digit placeholder we can choose a number from 0 to 9 independent of the other placeholders.
Say we have only one place holder. The number of possibilities is obviously 10.
Now we add another placeholder. We first put "0" for the first placeholder, and then run all the possible numbers for the second placeholder. We get 10 possible combinations.
Then we change the first placeholder to "1". We get 10 new combinations.
We can continue to change the first placeholder, and each time we get 10 new combinations.
Hence, for placeholders, we have different combinations.
If we add a third placeholder, we can take the previous 100 combinations and add another digit. So we now have possible combinations.
We see that the number of combinations is , where n is the number of placeholders.
We can prove it by induction. Assume this is true for n and we add the placeholder. Now the number of possible combinations is:

In the Social Security numbers there are placeholders, hence the number of possible combinations is:

There are 1 billion possible combinations.
In general we can say that if we have placeholders, each can hold set ...

Solution Summary

Discrete mathematics integers are examined. How many United States social security numbers are determined.

Please help with the following proofs. Answer true or false for each along with step by step proofs.
1) Prove that all integers a,b,p, with p>0 and q>0 that
((a+b) mod p)mod q = (a mod p) mod q + (b mod p) mod q
Or give a counterexample
2) prove for all integers a,b,p,q with p>0 and q>0 that
((a-b)mod p) mod q=0

Please see the attached file for the fully formatted problem.
Without using Theorem 4.2.2, use mathematical induction to prove that
P(n): 1 + 5 + 9 + ... + (4n - 3) = n(2n - 1) for all integers n >= 1

Please help with the following problem regarding discrete math.
I need a clear explanation of what an equivalence relation is with an examples. Specifically given 5|(m-n), where m and n are integers, please verify if this is an equivalence relation.
Please explain this clearly and in detail.

Please help me solve this equation! Can you please also show me the steps as to how to solve this so I know in the future? Thank you
I was unable to put lines between the #'s so it's supposed to be 2y over y - 1 = 4 over y + 9 - 7y over y squared - y. as written below. Thank you
2y/(y - 1) = (4/y) + (9 - 7y)/(y^2

Prove Each Directly.
1. The product of any two even integers is even.
Prove by cases, where n is an arbitrary integer and Ixl denotes the absolute
value of x.
2. [-x]=[x] (*Brackets are the x's is the absolute value symbol)
Give a counterexample to disprove each statement, where P(x) denotes an
arbitrary p

Proper walk through of following proofs required ( for a better understanding )
---
1) Prove that if n is an odd integer then n2 = 1 mod 8
2) Prove that 5n+3 is divisible by 4 for all integers n>=0

Let B = {0,{1},{2},{1,2}}; we define a relation on B. The pairs are of the form (X.Y) with X and Y subsets of {1,2}.
We set XRY if absX = absY.
Find the relations.
Please explain the answer clearly.

Please see the attached file for the full problem description.
The double bracket notation is pronounced " n multichoose k". The doubled parentheses remind us that we may include elements more than once.

1. Can Var(X_n) = (a^(2n - 2) + A^(2n - 4) + ... + A^2 + 1)sigma^2, be written as[ (sigma^2 )*(SUM(A2)n)from A=0 to A=n-1 =(sigma^2 )*(1/(1- A2 ) if n is large?
2. Why does Cov (Xi,Xj) = 0 if i /= j and sigma^2 if i=j
3. What is the basic equation for the Covariance?

1) Use Venn diagrams to determine whether each of the following is true or false:
a. (A union B) intersect C = A union (B intersect C)
b. A intersect (B union C) = (A intersect B) union (A intersect C)
2) Calculate the number of integers divisible by 4 between 50 and 500, inclusive.
3) Use the permutation formula to