M 2003
NUMBERS AND SETS – EXAMPLES SHEET 2
W. T. G.
1. Prove by induction that the following two statements are true for every positive inte
ger
n
.
(i) The number 2
n
+2
+ 3
2
n
+1
is a multiple of 7.
(ii) 1
3
+ 3
3
+ 5
3
+
. . .
+ (2
n

1)
3
=
n
2
(2
n
2

1)
.
2. Suppose that you have a 2
n
×
2
n
grid of squares (if
n
= 3 then you have a chessboard)
and you remove one square. Prove that, wherever the removed square is, the remaining
squares can be tiled with Lshaped tiles  that is, tiles consisting of three squares that form
a 2
×
2 grid with one square removed.
3.
By considering the equation (1

1)
n
= 0, give another proof that exactly half the
subsets of
{
1
,
2
, . . . , n
}
have even size.
4. Prove that
k
k
+
k
+ 1
k
+
. . .
+
n
+
k

1
k
=
n
+
k
k
+ 1
for any
n > k
. [Hint: for each set of size
k
+ 1 consider its largest element.]
5. Prove that
n
0
2
+
n
1
2
+
n
2
2
+
. . .
+
n
n
2
=
2
n
n
.
[Hint:
(
n
k
)
=
(
n
n

k
)
.]
6.
There are four primes between 0 and 10 and four between 10 and 20.
Does it ever
happen again that there are four primes between two consecutive multiples of 10?
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 Fall '10
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 Sets, Prime number, positive integers, positive integer, W. T. G

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