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

Statement 1:

5! And 6! Ends with one 0. So can be terminating or not terminating

Not sufficient.

Statement 2

S is odd. So S can be 3 or 5 which will lead to terminating or not terminating.

Not sufficient

Statement 1 + Statement 2

We get S is 5.

Sufficient.

OA should be C

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To determine whether 1/s is a terminating decimal, we need to consider the prime factorization of s. If s has prime factors other than 2 or 5, it will result in a repeating decimal.

Statement (1) tells us that s! ends with exactly one 0. This means s must have at least one factor of 5 in its prime factorization. However, we don't have information about other prime factors, so we cannot definitively determine whether 1/s is a terminating decimal. Statement (1) is not sufficient.

Statement (2) informs us that the sum of any two positive factors of s is even. This implies that s must be an even number without any odd prime factors. In other words, s can be written as 2^n * 5^m, where n and m are non-negative integers. Since s only contains the prime factors 2 and 5, 1/s will only have factors of 2 and 5 in its denominator. Therefore, 1/s will be a terminating decimal. Statement (2) is sufficient.

Considering both statements together, we have information about the prime factors of s. From statement (1), we know that s has at least one factor of 5. From statement (2), we know that s does not have any odd prime factors. Combining these two statements, we can conclude that s can be written as 2^n * 5^m, where n is a non-negative integer and m is a positive integer. Therefore, 1/s will have only factors of 2 and 5 in its denominator, making it a terminating decimal. Together, the statements are sufficient.

Therefore, the answer is (C) - both statements together are sufficient to answer the question.

1/s is terminating if s= 2^a 5^b, 1/(prime other than 2 or 5) is always non-terminating.
S1) [s/5]<1
=> 1<=s/5<2
=>s=5,6,7,8,9
1/5 is terminating, 1/6 is not (NS)

S2) Implies s is odd no.
S=5 (terminating) , s=7 (non-terminating)

S1+S2)
S=5 (terminating) , s=7 (non-terminating)
Not Suff

E)

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 no. is terminating if it can be expressed in a fraction of the form -> 1/2^n or 1/5^m or 1/2^n * 1/5^m.

(1) s! ends with exactly one is 0
-> s! ending with 1 zero implies 5<=s<=9, which implies we will have 3 in the denominator and therefore 1/s will not be terminating.
Sufficient.

(2) The sum of any two positive factors of s is even
-> 1 is a factor of all and for any two factors to have an even sum all of the other factors need to be odd. Therefore, s can be 3,5,7,11, etc.
if s is 5 then 1/s is terminating otherwise not.
Therefore since we don't have one answer, this statement is insufficient.

Ans=A.

IMO A

Tried to do this relatively quickly...

(1) - sufficient

for s! to have exactly 1 0 at the end, the min is 5! = 120 then did some math and it goes up to at least 8!, so clearly we need to use some kind of principle for this instead of brute force...

My thought was, any number that has a 3 as a factor will give a non-terminating decimal - tried a few...

1/3 = .33333
1/120 = .083333
So I figured as long as we have a 3, we will get a repeating situation, and since the min is 5! we have a 3 so this tells us 1/s is NOT a terminating decimal

(2) - insufficient
this would tell us s is odd. Why? 1 will be a factor of any positive integer, so we would need 1 + factor = even, so we need odd + odd, so it can only have odd factors. So we just need to find odd numbers that have terminating and non-terminating results of 1/s.

s = 3
1/3 = .3333

s= 5
1/5 = .2

insufficient!

statement 1:
s! ends in 1 zero i.e. s can be 5,6,7,8 or 9
1/5 can be terminating but 1/6 isn't
Not sufficient

statement 2:
even+even = even | odd+odd=even
since sum is even, only prime numbers can have that
lets take 3, factors are 1,3, sum is even
lets take 7, factors are 1,7, sum is even and so on
1/3 is non terminating...1/7 is non terminating
Sufficient

Answer is B

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. Insufficient
(2) The sum of any two positive factors of s is even. This means the number only has even factors, so any fraction formed with 1/s must be finite. Sufficient alone

Answer: (C)

Sort of hard to answer since the wording seems to be botched and is hard to comprehend, but assuming (i) is meant to read that there is s! ends with a singular "0," we have the following:

(i) Insufficient.
If s! ends in a singular 0, this could mean that s could take the following values: {5, 6, 9}. While 1/5 is a terminating decimal, neither 1/6 nor 1/9 are. Insufficient.

(ii) Insufficient.
Say, s = 3. Then, the positive factors are {1, 3}, which sum to an even number. Then, 1/3 is not a terminating decimal.
However, if s = 5, then the positive factors are {1, 5}, which also sum to an even number. 1/5 is a terminating decimal. There is more than one possibility, thus, insufficient.

(i) + (ii). Insufficient.
From (i) we know s can be {5, 6, 9}, and from (ii), s must be 5 or 9 (since 6 has factors that may not add to be even). However, 1/5 is a terminating decimal, while 1/9 is not. Thus, we have inconclusive results.

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

1/5! - non terminating decimal
1/6! - non terminating decimal
1/7! - non terminating decimal
1/8! - non terminating decimal
1/9! - non terminating decimal

NO is the answer

SUFFICIENT


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

If the sum of any two positive factors of s is even, then all the factors of s are odd

1/5! - non terminating decimal
1/15! - non terminating decimal
1/105! - non terminating decimal

NO is the answer

SUFFICIENT


(D) is the answer

A decimal terminates if the denominator has powers of 2 or 5 or both
\(s\) is a positive integer
Need to check if \(1/s\) is a terminating decimal

(1) s! ends with exactly one is 0

If \(s!\) ends in a single 0 then it would have only one factor of 5, hence possible values of \(s\) can be 5, 6, 7, 8, 9
We have the following
1/5 = 0.2 --> Valid
1/6 = 0.166666.... --> Invalid
1/7 = 0.142857.... --> Invalid
1/8 = 0.125 --> Valid
1/9 = 0.11111.... --> Invalid

Insufficient as we get both valid and invalid values

(2) The sum of any two positive factors of \(s\) is even
If the sum of positive factors is even then it should be an odd prime number
1 = 1 - Not Valid
2 = 1 and 2 - Not Valid
3 = 1 and 3 - Valid
4 = 1, 2 and 4 - Not Valid
5 = 1 and 5 - Valid
6 = 1, 2, 3 and 6 - Not Valid
7 = 1 and 7 - Valid
8 = 1, 2, 4 and 8 - Not Valid
9 = 1, 3 and 9 - Not Valid
10 = 1, 2, 5 and 10 - Not Valid
11 = 1 and 11 - Valid
In this scenario we can never have \(1/s\) as a terminating decimal if \(s\) is a prime number
Hence we get a clear No

Sufficient

Option B