Series Problem

Akansha wrote:If each term in the series a1 + a2 + … 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

OA is C

a1+a2+a3+a4........+an=350--------------1)

as here each term is either 7 or 77, i.e. a multiple of 7,
so above expression in 1) can be reduced to;
7(k1+k2+k3+k4+k5+.....kn)=350;
k1+k2+k3+k4+k5+.....kn=50; here k1,k2,k3,...kn is either 1 or 11;

now if we work with options, only option that will result in sum of 50 is C, and that when we have 39 1's and one 11. hence answer should be C

Another approach: x*77 + y*7=350 where x+y is our answer

We can write 350 in the following various pattern-
Since 77*5=385 so the max value of x can be 4.
4*77+6*7=350......(i)
3*77+11*7=350......(ii)
2*77+28*7=350......(iii)
1*77+39*7=350......(iv)

As per answer choices last pattern(iv) fits fine.(1+39=40)

Option:c

Great work, guys!

One thing that may be helpful - I saw this problem as a Units Digit Number Properties problem right away, and doing so allowed me to finish this thing in under ten seconds. As you're probably studying Number Properties quite a bit, look for opportunities to use them and you can save yourself a ton of time.

I saw it this way - the units digits that we're adding in each case:

77 + 7 + 7 + 7 + 77 + 7 .....

Are always 7s. And I need one of the answer choices to allow me to add up that number of 7s to get a units digit of 0.

Well, that means that the units digit is going to be formed by X * 7, and the only units digit of X that can multiply by 7 to form 0 is 0 itself. So 40*7 will end in a 0, and I knew that 40 was the only answer choice that would work.

Whether we're adding 77 or 7 is irrelevant - the 7 in the tens digit isn't going to affect the units digit. So this really comes down to just "what times 7 will give me a units digit of 0?", and if you see it that way the answer comes pretty quickly.

Incredibly fast trick Brain..!!! Also easy to get it. Thanks for sharing it.