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