“21 is a factor of
n
” and “
n
is a multiple of 42” only.
Since
n
is divisible
by 14 and 3,
n
contains the prime factors of both 14 and 3, which are 2, 7, and
3. Thus, any numbers that can be constructed using only these prime factors
(no additional factors) are factors of
n
. Since 12 = 2 × 2 × 3, you
cannot
make
12 by multiplying the prime factors of
n
(you would need one more 2).
However, you
can
construct 21 by multiplying two of the known prime
factors of
n
(7 × 3 = 21), so the second statement is true. Finally,
n
must be at
least 42 (= 2 × 7 × 3, the
least common multiple
of 14 and 3), so
n
is
definitely a multiple of 42. That is,
n
can only be 42, 84, 126, etc.
10.
6.
Start by considering integer
a
, which is the most constrained variable. It
is a positive one-digit number (between 1 and 9, inclusive) and it has four
positive factors. Prime numbers have exactly two positive factors. Prime
numbers have exactly two factors: themselves and one, so only look at non-
prime one-digit positive integers. That’s a short enough list:
1 has just one positive factor
4 has three positive factors: 1, 2, and 4
6 has four positive factors: 1, 2, 3, and 6
8 has four positive factors: 1, 2, 4, and 8
9 has three positive factors: 1, 3, and 9
So the two possibilities for
a
are 6 and 8. Now apply the two constraints for
b
.
It is 9 greater than
a
, and it has exactly four positive factors. Check the
possibilities:
If
a
= 6, then
b
= 15, which has four factors: 1, 3, 5, and 15.
If
a
= 8, then
b
= 17, which is prime, so it has only has two factors: 1 and
17.
Only
b
= 15 works, so
a
must be 6.
11.
(E).
Cutting a rectangular board into square pieces means that Ramon
needs to cut pieces that are equal in length and width. “Without wasting any
of the board” means that he needs to choose a side length that divides evenly
into both 18 and 30. “The least number of square pieces” means that he needs
to choose the largest possible squares. With these three stipulations, choose
the largest integer that divides evenly into 18 and 30, or the greatest common
factor, which is 6. This would give Ramon 3 pieces going one way and 5
pieces going the other. He would cut 3 × 5 = 15 squares of dimension 6” × 6”.
Note that this solution ignored squares with non-integer side length for the
sake of convenience, a potentially dangerous thing to do. (After all, identical
squares of 1.5” by 1.5” could be cut without wasting any of the board.)
However, to cut squares any larger than 6” × 6”, Ramon could only cut 2
squares of 9” or 1 square of 18” from the 18” dimension of the rectangle,
neither of which would evenly divide the 30” dimension of the rectangle. The
computed answer is correct.
12.
(B).
When dealing with remainder questions on the GRE, the best thing to
do is test a few real numbers:
Multiples of 6 are 0, 6, 12, 18, 24, 30, 36, etc.
Numbers with a remainder of 4 when divided by 6 are those 4 greater than the
multiples of 6:
x
could be 4, 10, 16, 22, 28, 34, 40, etc.
You could keep listing numbers, but this is probably enough to establish a
pattern.
(A)
→ ALL of the listed
x
values are divisible by 2. Eliminate (A).
(B)
→ NONE of the listed
x
values are divisible by 3, but continue
checking.
(C)
→ 28 is divisible by 7.
(D)
→ 22 is divisible by 11.
(E)
→ 34 is divisible by 17.
The question is “Each of the following could also be an integer EXCEPT.”
Since four of the choices could be integers, (B) must be the answer.
13.
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