5
7.
The prime numbers
p
and
q
are the smallest primes that differ by 6. What is the sum of
p
and
q
?
A
12
B
14
C
16
D
20
E
28
Solution:
C
Suppose
q
p
. Then
.
6
p
q
The prime numbers are 2, 3, 5, 7, …. . With
2
p
,
8
q
, which is
not prime. Similarly if
3
p
,
9
q
, which is also not prime. However, when
5
p
,
11
q
, which is
prime. So,
5
p
,
11
q
gives the smallest primes that differ by 6. Then
q
p
16
11
5
.
8.
Seb has been challenged to place the numbers 1 to 9
inclusive in the nine regions formed by the Olympic
rings so that there is exactly one number in each region
and the sum of the numbers in each ring is 11. The
diagram shows part of his solution.
What number goes in the region marked * ?
A
6
B
4
C
3
D
2
E
1
Solution:
A
We let
u
,
v
,
w
,
x
,
y
and
z
be the numbers in the regions shown. Since
the sum of the numbers in each ring is 11, we have, from the
leftmost ring, that
11
9
u
and so
2
u
. Then, from the next ring,
11
5
2
v
and so
4
v
. From the rightmost ring,
11
8
z
and so
3
z
.