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

Question

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.

limuengaoey · Answer

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

(1) a*b = 10^k, where k is a positive integer.
suppose a=5,b=2 then k=1
hence 5/2 is terminating decimal
suppose a=25, b= 4 then k=2
hence 25/4 is terminating decimal
hence 1 is sufficient

(2) b has three positive factors one of which is 5.
suppose b = 25 then the factors are 1,5,25
if a = 4 then a/b is the terminating factors.
if a = 11 then a/b is also terminating factors.
hence 2 is sufficient as well

The answer is D

missionmba2025 · Answer

(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

vedha0 · Answer

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

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

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