Question 168491: What's my telephone number? (_ _ _) _ _ _ - _ _ _ _
Each digit is different.
The Product of the sixth and seventh numbers is the third number.
The fourth, eigth, ninth, and tenth numbers are multiples of 3.
The sum of the fourth and sixth numbers is the same as the sum of the fifth and eigth numbers.
The second, third, sixth, and seventh numbers are powers of 2.
The first, fifth, seventh, and tenth numbers are prime.
THANK YOU SO MUCH!
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Note: I'm going to let a=first digit, b=second digit, c=third digit etc, until I get to j=tenth digit
So the telephone number looks like this:
(a b c) d e f - g h i j
Now let's do some translations:
"The Product of the sixth and seventh numbers is the third number. "
translates to
f*g=c
--------------------
"The fourth, eigth, ninth, and tenth numbers are multiples of 3"
translates to
d, h, i, j are multiples of 3 (giving us the possible values 0, 3, 6, 9)
Note: 0 is a multiple since 3*0=0
--------------------
"The sum of the fourth and sixth numbers is the same as the sum of the fifth and eigth numbers. "
translates to
d+f = e+h
--------------------
"The second, third, sixth, and seventh numbers are powers of 2."
translates to
b, c, f, g are powers of 2 which gives the possible values: 1, 2, 4, 8
Note: 1 is a power of 2 since 
--------------------
"The first, fifth, seventh, and tenth numbers are prime."
translates to
a, e, g, j are prime
List of (single digit) prime numbers: 2, 3, 5, 7
You will need to refer back to this list so it might help to write the list on a separate sheet of paper and cross out any digits that we've used
===============================================================
Note: remember, each digit is different which means that NO repeats are allowed.
Since digit j is both a multiple of 3 AND is prime, this means that the only possible digit for j is 3 (since 3 is both a multiple of 3 AND prime)
So j=3
The same can be done with g. Since g is both a power of 2 AND is prime, the only candidate is 2 (since 2 is both a power of 2 AND is prime)
So g=2
Looking at f*g=c, we know that all three are powers of 2. So this means that f*2=some power of 2
So let's try some values of f
f=1 (CANNOT do since 1*2=2; a repeat would occur where g=c)
f=2 (CANNOT do since 2 is already taken)
f=4 (CAN do since 4*2=8; all three are powers of 2)
f=8 (CANNOT do since 8*2=16; 16 is comprised of 2 digits)
So f=4 and c=8
Since we know that g=2, f=4, and c=8 (and there are ONLY 4 digits that are powers of 2), this means that b=1 (by elimination)
Just to recap, we've taken the digits 3, 2, 4, 8, and 1 so far (5 digits taken already).
--------------------------------------
Now let's move onto the equation
d+f = e+h
We know that f=4, so let's plug that in:
d+4 = e+h
Here's where we have to try some combinations
The possible values for d are 0, 6, and 9 (remember d is a multiple of 3 and the digit 3 is already taken)
The possible values for e are 5 and 7 (e is prime and the digits 2 and 3 are already taken)
The possible values for h are 0, 6, and 9 (remember h is a multiple of 3 and the digit 3 is already taken)
So we could have the following:
d=0 OR d=6 OR d=9
e=5 OR e=7
h=0 OR h=6 OR h=9
So let's add some combinations:
Combination 1:
If d=0, e=5, and h=6 (0 already taken), then
0+4=5+6
4=11 ... which is false (cross this combination out)
Combination 2:
If d=0, e=5, and h=9, then
0+4=5+9
4=14 ... which is false (cross this combination out)
Combination 3:
If d=0, e=7, and h=6 (0 already taken), then
0+4=7+6
4=13 ... which is false (cross this combination out)
Combination 4:
If d=0, e=7, and h=9, then
0+4=7+9
4=16 ... which is false (cross this combination out)
Combination 4:
If d=6, e=5, and h=0 (6 already taken), then
6+4=5+0
10=5 ... which is false (cross this combination out)
Combination 5:
If d=6, e=5, and h=9 (6 already taken), then
6+4=5+6
10=11 ... which is false (cross this combination out)
Combination 6:
If d=6, e=7, and h=0 (6 already taken), then
6+4=7+0
10=7 ... which is false (cross this combination out)
Combination 7:
If d=6, e=7, and h=9 (6 already taken), then
6+4=7+9
10=16 ... which is false (cross this combination out)
Combination 8:
If d=9, e=7, and h=0 (9 already taken), then
9+4=7+0
13=7 ... which is false (cross this combination out)
Combination 9:
If d=9, e=7, and h=6 (9 already taken), then
9+4=7+6
13=13 ... which works
So after trying 8 combinations of numbers, we get
d=9, e=7 and h=6
-----------------------------------------------------------
So to recap, we've taken the digits 3, 2, 4, 1, and 8 (from the first part) and the digits 9, 7, and 6 (from the second part)
Altogether, we know that
j=3, g=2, f=4, b=1, and c=8 (from part 1)
d=9, e=7 and h=6 (from part 2)
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Since d=9, h=6, and j=3, using elimination we get i=0 (note: I'm looking at the "multiples of 3" group)
Finally, since e=7, g=2, and j=3, this means that a=5 (note: I'm looking at the "prime numbers" group)
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Answer:
So we have the following numbers:
a=5
b=1
c=8
d=9
e=7
f=4
g=2
h=6
i=0
j=3
which means that the telephone number is
(518) 974-2603
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