Question

When a certain number is multiplied by 13, the product consists entirely of fives. The smallest such number is _ _ _.
A. 41625
B. 42515
C. 42735
D. 42135

Answer 1

Hint:- As we know that whenever we do the product of quotient and divisor, we will simply get the dividend. Here we know that the dividend must be the number of fives and divisor is 13.

Complete step-by-step answer:
First of all let us assume the number which when multiplied by 13 gives the product consisting entirely of fives as ‘x’.
So, the number = x
Now the equation must be like that the multiplication of x and 13 will give a number with only 5.
So, x*13 = number with only fives
Now on further solving it we will get x in terms of number with only five and 13
So, x = \[\dfrac{{{\text{Number with all digit as 5}}}}{{13}}\]
Now to make this question more understandable to us we are considering some cases.
In these cases we will only consider the values with only 5.
Case 1:- Let us assume that the number with only five = 555
As we know from above that,
x = \[\dfrac{{{\text{Number with all digit as 5}}}}{{13}}\]
So, x = \[\dfrac{{555}}{{13}}\]
x = 42.69
As 555 is not completely divisible by 13. Hence, this value (i.e. 555) is absolutely wrong.
Case 2:- Let us assume that the number with only 5 is 55555
Similarly from the above formula we know that x must be equal to the completely divisible value of only five by 13.
Now putting the value in the above equation. We get,
x = \[\dfrac{{55555}}{{13}}\]
x = 4273.46
So, 55555 is also not completely divisible by 13. So, this will be a wrong answer.
Case 3:- Now let us assume the number with only five = 555555
And as above mentioned
x = \[\dfrac{{{\text{Number with all digit as 5}}}}{{13}}\]
x = \[\dfrac{{555555}}{{13}}\]
x = 42735
In this case the number only with five (i.e. 555555) is completely divisible by 13. So, the smallest number which when multiplied by 13 gives the product number completely with fives is 42735.
Hence, the correct option will be C.

Note:- Whenever we come up with this type of problem then we have to find the answer of the problem by hit and trial method. In this question we had to find a number exactly divisible by 13 and consisting all digits as 555555. So, dividing 555555 by 13 we get 42735 as a quotient. So, we had to only check the condition for different numbers like 555, 5555, 55555 and 555555 etc. And the first number which satisfies the condition will be the required answer.