CMSC250, Spring 2004 Homework 6
Due Wednesday, March 10 at the beginning of your discussion section.
You must write the solutions to the problems single-sided on your own lined
paper, with all sheets stapled together, and with all answers written in sequential
order or you will lose points.
1. Let p1, p2, . . . , pnbe distinct prime numbers such that p1= 2 and n > 1, and let
a=p1p2···pn. Prove that a≡42.
Hint: Use the quotient-remainder theorem, and eliminate the other possibilities.
2. An integer is called squarefree if it is not divisible by the square of any prime number.
What can you deduce about the standard factored form of a squarefree number?
Hint: Examine the exponents.
3. Prove that any integer a > 1 can be written as either a perfect square, a squarefree
integer, or the product of a squarefree integer and a perfect square.
Hint: Break this into the three cases. Two of them should be easy. For the third case,
write aas (pe1
1pe2
2···pen
n) (qd1
1qd2
2···qdm
m) where all the ei’s are even and all the dj’s are
odd.
4. Prove that for all real numbers x,x−1<bxc ≤ x.
5. Write the first four terms of each of the following sequences:
(a) ∀j > 1, sj=j(j−1)
(b) ∀n≥0, rn=n
n!
6. Reduce each of the following expressions to a single numeric value:
(a)
4
X
i=1
−1
i2+i
(b)
10
Y
h=0
h
h! + h+ 1
(c)
3
X
m=1 m
Y
n=1
m
n!
7. Write each of the following using sum and/or product notation:
(a) n+ 0 −n−1 + n+ 2 −n−3 + n+ 4
(b) √5(1) + 2(1 + 2) + √3(1+2+3)+√2(1+2+3+4)+1(1+2+3+4+5)
1