SAT Math Hard Practice Quiz Answers
7.
60
(Estimated Difficulty Level: 5)
First, note that 15
mn
= 3
·
5
·
mn
. We need
√
3
·
5
·
mn
to be an integer. We could have, for example,
m
= 3
and
n
= 5 since
√
3
·
5
·
3
·
5 =
√
3
2
·
5
2
= 15, except
that the problem requires that
mn >
15. (This is a hard
problem for a reason, after all!) If
m
= 3
·
2 and
n
= 5
·
2
then 15
mn
= (3
·
5)(3
·
2)(5
·
2) = 3
2
·
5
2
·
2
2
. That way,
√
15
mn
=
√
3
2
·
5
2
·
2
2
= 30 is still an integer, making
the least possible value of
mn
equal to 6
·
10 = 60.
8.
B
(Estimated Difficulty Level: 4)
A good opportunity to plug in real numbers! For ex-
ample, suppose set
M
consists of the integers: 2, 4, 6,
8, 10, and 12. The sum of the least three is 12 and the
sum of the greatest three is 30, so answer B is correct.
You say you want an algebraic solution? Suppose that
n
is the first even integer. The remaining integers are then
n
+2,
n
+4,
n
+6,
n
+8, and
n
+10. The sum of the least
three of these integers is
x
=
n
+(
n
+2)+(
n
+4) = 3
n
+6,
and the sum of the greatest three of these integers is
y
= (
n
+6)+(
n
+8)+(
n
+10) = 3
n
+24. So,
y
−
x
= 18,
or
y
=
x
+ 18.
9.
792
(Estimated Difficulty Level: 5)
To get the greatest difference, we want to subtract a
small number from a large one, so we will need the
digit 9 and the digit 1, in order to make a number in
the 100’s and a number in the 900’s. The large number
will look like 9
N
1 and the small number will look like
1
N
9, where
N
is a digit from 2 to 8. You will find that,
no matter what you make
N
, the difference is 792.
10.
E
(Estimated Difficulty Level: 5)
Suppose that the integer is
n
. The result of subtract-
ing
n
from its square is
n
2
−
n
=
n
(
n
−
1), which is
the product of two consecutive integers, so answer E is
correct.
Notice that if you multiply any two consecutive integers,
the result is always even, since it is the product of an
even integer and an odd integer. To win an Erik The
Red Viking Hat, see if you can determine why the result
is never a negative integer.
erikthered.com/tutor
pg. 11