Around the World in 80 Questions (Day 10): If a and b are positive

(1) a*b = 10^k, where k is a positive integer.

a and b consists of only 2 and 5 as its prime factors. Hence, we can conclude that a/b will always be a terminating decimal.

Sufficient.

(2) b has three positive factors one of which is 5.

Three positive factors indicate that the number b is a perfect square. Hence, b = 25. As the denominator consists of only 5s. We can conclude that a/b will always be a terminating decimal.

Sufficient.

IMO D

a/b are +ve integers target is a/b a terminating decimal

for a terminating decimal the dr has to be in form of 2^a *5^b or any one of these factors only
#1
a*b = 10^k, where k is a positive integer.
a*b= 2^k * 5^k
2^k/5^k and
5^k/2^k
will always be a terminating decimal
sufficient
#2
b has three positive factors one of which is 5.
b = 5^2
this will be sufficient
or
b=5 * 3*1 or 5*3*7
insufficient
option A is correct

\(\frac{a}{b } \) is non-terminating in case b has prime as a factor when \frac{a}{b} is is it's lowest form

A fraction in lowest terms can be expressed as a terminating decimal if and only if its denominator has no prime factors other than 2 or 5

S1) \(ab=2^x 5^y\)
denominator has factor of 2 or 5 or both

(Suff)

S2) b could have a prime number *5
b=10 (terminating)
b=15(non-terminating)

(NS)

Hence A)

1. a*b = 10^k means that the product of a and b is a power of 10. This implies that a and b could be 2 and 5 in any combination. For example, if k=2, then a*b = 100, and possible pairs for (a, b) could be (1, 100), (2, 50), (4, 25), (5, 20), (10, 10), etc.
However, it's also possible that a or b could be a number that includes other prime factors. For example, in the pair (4, 25), 4 is not a prime number and includes the prime factor 2 twice. Therefore, statement (1) alone is not sufficient to determine whether a/b is a terminating decimal.

2. b has three positive factors one of which is 5. This implies that b could be 5, 25, or 5 times a prime number other than 2. In the first two cases, a/b would be a terminating decimal, but in the third case, it would not. So, this statement is also not sufficient.

From statement (1), we know that a and b are composed of the prime factors 2 and/or 5.
From statement (2), we know that b has three positive factors, one of which is 5. This means that b could be 5 (with factors 1, 5), 25 (with factors 1, 5, 25), or 5 times a prime number other than 2.
However, since we know from statement (1) that b can only have the prime factors 2 and/or 5, the third option for b is ruled out. Therefore, b must be either 5 or 25, both of which would result in a/b being a terminating decimal.
So, the two statements together are sufficient to answer the question. Hence C

If a and b are positive integers, is a/b a terminating decimal ?

(1) a*b = 10^k, where k is a positive integer.

(2) b has three positive factors one of which is 5.
______________________________________
(1) if that is the case, one of a or b is multiple of 10. So, it is sufficient.
(2) if one of factors is five, we can answer the question. Moreover, other factors are 1 and the number itself B. It is still enough to answer - Sufficient.

D is our winner.

If a and b are positive integers, is a/b a terminating decimal ?

(1) a*b = 10^k, where k is a positive integer.

(2) b has three positive factors one of which is 5.


Case I: a*b = 10^k where k is a positive integer
Prime factors of 10 is 5 and 2
K cannot be 0 as 0 is neither positive nor negative
so a or b can be one of the powers of 2 or 5
and we know that power of 2 or 5 make fraction a terminating decimal
so this option is sufficient

Case II: b has three positive factors one of which is 5
lets say b has 2,3 and 5 , then it will be non terminating decimal
but if b has 1, 2, 5 then it will be terminating decimal
or b has 1, 3, and 5 then it will be non terminating decimal
so this option is not sufficient

A is the answer

The Answer is C
We have been given that a and b are positive integers and we need to find if a/b is terminating decimal.
Since a terminating decimal must have either 2 or 5 or both as its prime factors in the denominator, then the decimal will terminate.
In Statement A, we get the information that a x b=10K. Since we don't know if a=2 or 1 and b= 5,2 ,1 or 10 we cannot say for sure if a/b would terminate. Hence insufficient.
In Statement B we find out that b is not a prime number. Hence this statement is insufficient on its own.
Combining both, We know that a x b is equal to some power of 10. Through B we know that it is not a prime. We can deduce that b =10 and a =1. Hence it is a terminating decimal.
C is sufficient

If a and b are positive integers, is a/b a terminating decimal ?

(1) a*b = 10^k, where k is a positive integer.
This is sufficient.
4 x 25 will result in a terminating decimal
10 x 100 will result in a terminating decimal
5 x 20 will result in a terminating decimal


(2) b has three positive factors one of which is 5.
This is sufficient.
b = 25. Thus any number divided by 25 will result in a terminating decimal
Answer = D

(1) a*b = 10^k, where k is a positive integer.
That means ab = (5*2)^k
That means b will have some power of 5 or 2 or both and that is enough to declare that a/b is terminating. Sufficient.

(2) b has three positive factors one of which is 5.
The factor can be 3, 7 19. We have no way of knowing. Insufficient.

Thus A

S1:- \(a*b = 10^k = 5^k * 2^k\)

Possible values of a and b will be multiples of \(1^k and 10^k or 5^k * 2^k\)

So, for each multiple of 2 or 5 in the denominator it will create a terminating decimal. So, S1 is sufficient.

S2:- Let other factors = 3 and 7, then a/b will not be a terminating decimal.
We can also chose values of other two factors such that a/b will be terminating factor. So, S2 not sufficient.

Correct choice is A.

Statement (1) tells us that the product of 'a' and 'b' is equal to 10^k, where k is a positive integer. However, we don't have any information about the individual values of 'a' or 'b', so we can't determine if a/b is a terminating decimal. Statement (1) is insufficient.

Statement (2) states that 'b' has three positive factors, and one of them is 5. This implies that 'b' can be expressed as 5 times a positive integer. However, without knowing the value of 'a', we can't determine if a/b is a terminating decimal. If 'a' is a multiple of 5, then a/b is terminating, but if 'a' is not a multiple of 5, then a/b is non-terminating. Statement (2) is insufficient.

Combining both statements doesn't provide any new information about 'a' or 'b', so we still can't determine if a/b is a terminating decimal.

Therefore, the answer is (E) Both statements together are insufficient to determine if a/b is a terminating decimal.

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(1) a*b = 10^k, where k is a positive integer.
This means a and b are exponents of 2,5 or 10. Does not matter which of them are a and b, it will always give a terminating decimal for any arrangement of a and b. Sufficient.

(2) b has three positive factors one of which is 5.
This does not give any information about a. Also, information about b is also partial. We do not know the value of b as such. Hence insuffcient.

A is the correct answer.

To determine the fraction is terminating or not, we need to check the denominator. First express the fraction in lowest form, and write the prime factor in denominator. If denominator has the only 2 and 5 as prime factor. It is terminating and if it has other prime factors in it then it is non-terminating.

Here, st1. says nothing about the denominator. Insufficient

St2. confirm that there is three positive factors and one of them is 5 but still don't know about the other two. Insufficient

Combining both as well, we can't confirm, denominator has only '2 and 5' prime factors.

E

statement 1
lets assume k=2
a*b=100
a ,b can be 5,20 or 25/4 etc
and it will be always a terminating decimal for any value of k , a ,b
statement 1 is sufficient
statement 2
only square of a prime has 3 factors
b is 25
any no divided by 25 will be terminating decimal
statement 2 is also sufficient
answer is d

answer is D.

a and b are positive integers. for a/b to give a terminating decimal representation, the denominator should be of the form 2^m * 5 ^n

STATMENT 1 : a*b = 10^k
10^k = (2^k)*(5^k). so a and b will be of the form (2^m)*(5^n) where 0 <= m,n <= k and the addition of powers from a and b will be k. anyways, what we require is that b will be of the form (2^m)*(5^n). therefore a/b will have terminating decimal. so its sufficient

STATEMENT 2 : b has 3 +ve factors, one of which is 5
Note that for any positive integer, 1 and the integer itself will be factors. so using this and the given statement, factors of b are 1,5,b. so 1*b = 5*5. thus b will be 25. which is 5^2, that is no of factors is 3.
now that b=25 is sure, a/b will have terminating decimal. so its sufficient

EACH STATEMENT IS SUFFICIENT. ANSWER IS D

Answer is A
If a and b are positive integers, is a/b a terminating decimal ?
Question is asking is b equals to 2^x * 5 ^y ?
(1) a*b = 10^k, where k is a positive integer.
Sufficient. a and b has only 2 prime factors which are 2 and 5.
(2) b has three positive factors one of which is 5.
b has other factors other than 2 and 5. we don't know of. Insufficient.

For a/b to be a terminating decimal, the simplified denominator must only have factors 2 or 5 or both

(1) a*b=10^k -> a, b have factors 2 and 5 only -> Sufficient
(2) b has 3 positive factors -> b=k^2 (k is prime). Since b has a factor 5 -> k=5 -> b=5^2 -> Sufficient

Answer D