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(1) Consider the binomial expansion (
a
+
b
)
n
=
C
n
0
a
n
+
C
n
1
a
n

1
b
+
...
+
C
n
n
b
n
.
Show that
C
n
0

C
n
1
+
C
n
2

...
±
C
n
n
= 0
Hint:
Pick appropriate
a
and
b
.
(2) Find all prime numbers smaller than 100.
(3) Recall that prime twins are pairs of prime numbers of the form
n,n
+
2 such as 3 and 5 or 11 and 13.
Similarly, prime ”triplets” are triples of prime numbers of the form
n,n
+ 2
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Unformatted text preview: ,n + 4. Find all prime triplets. Hint: Show that one of the numbers n,n +2 ,n +4 must be divisible by 3. (4) Show that there is no natural number k such that 2 k 1( mod 6). Find all possible values of 2 k ( mod 6). (5) Prove that for any natural k 4 k + 4 9 k 0( mod 5) 1...
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This note was uploaded on 04/26/2011 for the course MATH 246 taught by Professor Applebaugh during the Spring '10 term at University of Toronto Toronto.
 Spring '10
 Applebaugh
 Math, Prime Numbers, Binomial

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