12 Days of Christmas GMAT Competition - Day 10: Any decimal that has

Question

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Any decimal that has only a finite number of nonzero digits is a terminating decimal. For example, 24, 0.82, and 5.096 are three terminating decimals. If s is a positive integer and the ratio 1/s is expressed as a decimal, is 1/s a terminating decimal?

(1) s! ends with exactly one is 0
(2) The sum of any two positive factors of s is even

Archit3110 · Answer

given that s is a +ve integer , target is 1/s terminating decimal

#1
 s! ends with exactly one is 0
s can be 5,6,7,8,9
we get both yes an no to target

#2
The sum of any two positive factors of s is even
s can be 3,5,7,9
insufficient 
from 1 &2
s can be 5 or 7
  option E

missionmba2025 · Answer

1/s is a terminating decimal when s consists only integral powers of 2 and 5. (1) s! ends with exactly one is 0 As s! ends with one zero 4 < s < 10 s = 5 => 1/s is terminating s = 7 => 1/s is not terminating No sufficient. (2) The sum of any two positive factors of s is even The sum of any two positive factors of s is even => s is a prime number greater than 2 s = 5 => 1/s is terminating s = 7 => 1/s is not terminating No sufficient. (1) + (2) s can still be 5 or 7. Not sufficient. IMO E

VivekSri · Answer

I will go with option E.

(1) s! ends with exactly one is 0

5! = 120, 6! = 720,
7! = 5040, 8! = 40320 and 9! = 362880,

Now 1/5= terminating 1/6 = Non terminating, so not sufficient

(2) The sum of any two positive factors of s is even

This clearly means number is odd as any two factor, as one factor of any number is one, in order to make it even the other factor has to be odd, so let us take a example 5 = factor (1,5) sum is even and 1/5 = terminating, 7 = factor(1,7) sum of factor is even, but 1/7 is non terminating, so not sufficient.

Combining both :

5! = 120, 7! = 5040, 9! = 362880.

Not Possible also 1/5 = Terminating but 1/7 is non terminating.

So option E

Bunuel · Answer

GMAT Club Official Explanation:

Any decimal that has only a finite number of nonzero digits is a terminating decimal. For example, 24, 0.82, and 5.096 are three terminating decimals. If s is a positive integer and the ratio 1/s is expressed as a decimal, is 1/s a terminating decimal?

A reduced fraction  \frac{a}{b}  (meaning that the fraction is already in its simplest form, so reduced to its lowest term) can be expressed as a terminating decimal if and only if the denominator  b  is of the form  2^n5^m , where  m  and  n  are non-negative integers. For example:  \frac{7}{250}  is a terminating decimal  0.028 , as the denominator  250  equals  2*5^3 . The fraction  \frac{3}{30}  is also a terminating decimal, as  \frac{3}{30}=\frac{1}{10}  and the denominator  10=2*5 .
 
 Note that if the denominator already consists of only 2s and/or 5s, then it doesn't matter whether the fraction is reduced or not. 
 
For example,  \frac{x}{2^n5^m} , (where  x ,  n , and  m  are integers) will always be a terminating decimal.
 
(We need to reduce the fraction in case the denominator has a prime other than 2 or 5, to see whether it can be reduced. For example, the fraction  \frac{6}{15}  has 3 as a prime in the denominator, and we need to know if it can be reduced.)
 
(1) s! ends with exactly one is 0

This implies that s! has only one factor of 5, hence s can be 5, 6, 7, 8, or 9. If s is 5 or 8, 1/s will be a terminating decimal. However, if s is 6, 7, or 9, 1/s will NOT be a terminating decimal. Not sufficient.

(2) The sum of any two positive factors of s is even

This implies that all factors of s are odd, so s itself is odd. If s is a power of 5: 5, 25, ... and so on, 1/s will be a terminating decimal. However, if s is any other odd number (greater than 1), 1/s will NOT be a terminating decimal. Not sufficient.

(1)+(2) Since from (2) s is odd, then from (1) it can be 5, 7, or 9. If s is 5, 1/s will be a terminating decimal. However, if s is 7, or 9, 1/s will NOT be a terminating decimal. Not sufficient.

Answer: E.

12 Days of Christmas GMAT Competition - Day 10: Any decimal that has

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