T is the set of all 4-digit integers in which each digit is

Question

T is the set of all 4-digit integers in which each digit is odd. What is the remainder when the 94th smallest element of T is divided by 25?

A) 1
B) 2
C) 4
D) 9
E) 22

Va16 · Answer

Since we need to find the remainder, we have to know what the last digit of the 94th smallest element is, which could be 1, 3, 5, 7 or 9. 
These digits will appear in the list of terms in a cyclical manner, so we can find the remainder when 94 is divided by 5 (the number of odd numbers) which will tell us which number is the last digit.

Since the remainder is 4, 7 is the last digit, and the remainder is 2.

Is this a reliable approach?

Adit_ · Answer

What if I have 57 as the last two digits of the number? The remainder is 7 here.
So finding the last digit alone is not enough.

For divisibility by 25 you only need the last two digits.
Like you pointed out it is cyclic.
94/25 we have remainder as 19 here
Which is completing 3 cycles with 1,3,5 as tens digit and then 4th cycle's second term is considered.
Which is 7 followed by the second last digit in sequence = 7
Thus last 2 digits = 77
Remainder = 77 mod 25 = 2.

Va16 · Answer

Thanks for the help! For some reason I had read 5 instead of 25.

pappal · Answer

smallest no. is 1111 and only changes at all the places can be 1,3,5,7,9 (odd).
i.e. for 111 as left 3 digits -there are 5 choices for the units place or 5 possible numbers.
1111,1113,1115,1117,1119
 similarly for 11 as left 2 digits - there are 5*5=25 possible numbers.
it means that for 13 as left 2 digits - there are 5*5=25 possible numbers.(totaling-50) 
and for 15 as left 2 digits - there are 5*5=25 possible numbers.(totaling-75)
now we are for sure that our 94th smallest no. has 17 as first two digits; 
and for 1 at tens place-5 numbers; for 3 at tens place- next 5 numbers; for 5 at tens place- next 5 numbers. (totaling 90).
so the four next numbers should be having 7 at the tens place--1771,1773,1775,   1777 
 Remainder of(1777/25)=2
B

HarshavardhanR · Answer

What a fun question!!

The smallest number -> 1111
The next numbers -> 1113, 1115, 1117, 1119, then, 1131, 1133, etc., till 9999.

The total number of numbers in this set - 5 options for the left most digit x 5 options for the next digit x 5 options for the tens digit x 5 options for the ones digit = 625. We need to find the remainder when the 94th number in this set is divided by 25.

We can arrive at the 94th number by  moving in ascending order  in the following manner ->

(1) 11 _ _

There are 5 x 5 = 25 such numbers. Total - 25 numbers counted so far (in ascending order).

(2) 13_ _

This is another 5 x 5 = 25 numbers. Total - 50 numbers.

(3) 15 _ _

This is another 5 x 5 = 25 numbers. Total - 75 numbers.

Now, things get interesting. If we got for 17 _ _, that would be another 5 x 5 = 25 numbers, which would lead to a total of 100 numbers. Hence, the 94th number of the set is somewhere in between.

So, let's split this group into more categories, based on ten's digit.

(4) 171_

5 such numbers (corresponding to 1711, 1713, 1715, 1717, 1719). Total - 80

(5) 173_ 
In a similar fashion, there a 5 numbers here. Total - 85

(6) 175_

Another 5 numbers. Total - 90

(7) 1771, 1773, 1775 --- are the 91st, 92nd, and 93rd numbers.

So,

The 94th number of the set = 1777.

The remainder when 1777 is divided by 25 = 2.  Choice B .

T is the set of all 4-digit integers in which each digit is

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T is the set of all 4-digit integers in which each digit is

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