Problem Solving

teejaycrown wrote:If each term in the sum a1+a2+a3........+an is either 7 or 77 and the sum equals 350 which of the following could be equal to n?

a. 38
b. 39
c. 40
d. 41
e. 42

Let x and y be the number of 7's and the number of 77's respectively.
Then 7x + 77y = 350 implies x + 11y = 50
x + y = 50 - 10y
So, x + y = n = 50 - 10y implies n must have the unit digit of 0 because 50 - 10y must be a number which has units digit as 0. C is the only option in which the units digit is 0.

The correct answer is C.

teejaycrown wrote:If each term in the sum a1+a2+a3........+an is either 7 or 77 and the sum equals 350 which of the following could be equal to n?

a. 38
b. 39
c. 40
d. 41
e. 42

Looking for a pattern is one possible route:

Since both 7 and 77 have 7 as their units digit, we know that if we take any two terms, their sum will have a units digit of 4 (e.g., 7 + 7 = 14, 7 + 77 = 84, 77 + 77 = 154)

Similarly, if we take any three terms, their sum will have a units digit of 1. (e.g., 7 + 7 + 7 = 21, 7 + 7 + 77 = 91, etc.)

Now let's look for a pattern.

The sum of any 1 term will have units digit 7
The sum of any 2 terms will have units digit 4
The sum of any 3 terms will have units digit 1
The sum of any 4 terms will have units digit 8
The sum of any 5 terms will have units digit 5
The sum of any 6 terms will have units digit 2
The sum of any 7 terms will have units digit 9
The sum of any 8 terms will have units digit 6
The sum of any 9 terms will have units digit 3
The sum of any 10 terms will have units digit 0
The sum of any 11 terms will have units digit 7 (at this point, the pattern repeats)

From this, we can conclude that the sum of any 20 terms will have units digit 0
And the sum of any 30 terms will have units digit 0, and so on.

We are told the sum of the terms is 350 (units digit 0), so the number of terms must be 10 or 20 or 30 or . . .

Since C is a multiple of 10, this must be the correct answer.

Cheers,
Brent

Another option is to try some different configurations. If we do, we see that adding 39 7's and 1 77 gives us a total of 350. (39 + 1 = [spoiler]40 = E[/spoiler])