If n th date of August and 2n th date of October of the same year fall

Question

If  n_{th}  date of August and  2n_{th}  date of October of the same year fall on the same day of the week, then how many values of n are possible ?

A. 0
B. 1
C. 2
D. 3
E. 4

chetan2u · Answer

August consists of 31 days and september consists of 30 days so, total difference is 31+30=61 days..
So the 1st August and 2nth or 2nd oct will have a difference of 61+1.
Thus difference between nth august and 2nth October is 61+n..

For  n_{th}  date of August and  2n_{th}  date of October of the same year to fall on the same day of the week, 61+n should be a multiple of 7 and  2n\leq{30} 
61+n...
when n=2..61+3=63=7*9..yes
add another 7....n=2+7=9....61+2+7=61+9=70..Yes
add another 7..n=2+7+7=16......61+2+7+7=61+16=77..Yes, but 2n=2*16=32>30..so discard

Hence, two values n=2, and n=9

C

Ansh777 · Answer

Lets consider nth day in August. 
No of day still remaining in August= 31-n
No of day in Sep= 30
Now if we are at 2nth day in Oct, No of days in Oct= 2n also 2n<31

Therefore from nth August to 2nth Oct, Total No of days= 31-n+30+2n = (61+n)

And since n & 2n lie on same day of the week, so (61+n) must a multiple of 7.

Now possible values of n such that (61+n) is a multiple of 7 are:
n=2; 61+n= 63 which is a multiple of 7
n=9; 61+n= 70 which is a multiple of 7
n=16; 61+n= 77 which is a multiple of 7; but if n=16 2n=32 which is not possible.
Therefore only possible values of n are 2 and 9

Answer: C

Franzhel · Answer

I used a simpler approach to have a good enough estimation on the answer.

Since you have around 30 days in a month, n should be limited to 15 because you have 2n in October. Then if you have 15 days, you should have the days repeat themselves twice. Lets just assume, to simplify, that both months start with a Monday on the 1st. Then the 1st of August will be a monday and the 2nd of August will be a Tuesday. Then, the 2nd of august will be a Tuesday and the 4th of august will be a Thursday. If you follow this, the 7th of August will be a Sunday and the 14th of Oct will be a sunday too. This will repeat itself with the 14th of August and the 28 of October.

However, if you realize that you have 15 days available and that each value of n in August will skip a day in October you can realize that the answer will be 2.

Therefore, answer should be C (2)

bumpbot · Answer

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If n th date of August and 2n th date of October of the same year fall

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If n th date of August and 2n th date of October of the same year fall

Prep Toolkit

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