Answer
Verified
447.6k+ views
Hint: We need to find the value of digit d in the eight-digit number 2575d568. As we know that 2575d568 is divisible by 54 and 87. So, we can say that it is also divisible by the factors of 54 and 87. So, we need to find the factors of 54 and 87 first and then with the help of the divisibility rule for each factor, find the value of digit d.
Complete step by step answer:
By the prime-factorization method, we can find the factors of 54 and 87. So, we have:
Factors of 54 are: \[54=2\times 3\times 3\times 3=2\times 27\]
Factors of 87 are: $87=3\times 29$
So, the number 2575d568 to be divisible by 54 and 87, it should be divisible by 2, 27, and 29.
Since the unit’s digit of the number is an even number, therefore 2575d568 is always divisible by 2.
Since both 54 and 87 have 3 as their common factor. So, the eight-digit number 2575d568 must be at least divisible by 3.
To check whether a number is divisible by 3 we need to follow the divisibility rule by 3. It says that a number is divisible by 3 when the sum of digits is divisible by 3.
Example: $6561\Rightarrow 6+5+6+1=18$ is divisible by 3.
So, to find whether 2575d568 is divisible by 3 or not we need to check if the sum of digits of the number is divisible by 3 or not.
Therefore, we can write:
$\begin{align}
& \Rightarrow 2+5+7+5+d+5+6+8 \\
& \Rightarrow 38+d \\
\end{align}$
So, we need to put a one-digit value of d that makes $38+d$ divisible by 3.
i.e. d = 1, 4 or 7
If we add d=1, 4 or 7 in $38+d$ we get 39, 42 or 45 which are divisible by 3.
So, for d=1, 4 or 7, the number 2575d568 is divisible by 3. Hence the number would be divisible by 27.
But we need to check if for d=1, 4 or 7 the number 2575d568 is divisible by 29.
Put d=1, 4 or 7 in 2575d568, and check whether the number is divisible by 29. We get to know that for d=7, the number 2575d568 is divisible by 29.
Hence for d=7, the number 2575d568 is divisible by 54 and 87 both.
So, the correct answer is “Option b”.
Note: While checking that if the number 2575d568 is divisible by 3, we get three values of d = 1, 4, and 7. Don’t assume that the number will be divisible by 54 and 87 deliberately. The first check for other factors, i.e. 27 and 29 then confirm if it is divisible or not.
Complete step by step answer:
By the prime-factorization method, we can find the factors of 54 and 87. So, we have:
Factors of 54 are: \[54=2\times 3\times 3\times 3=2\times 27\]
Factors of 87 are: $87=3\times 29$
So, the number 2575d568 to be divisible by 54 and 87, it should be divisible by 2, 27, and 29.
Since the unit’s digit of the number is an even number, therefore 2575d568 is always divisible by 2.
Since both 54 and 87 have 3 as their common factor. So, the eight-digit number 2575d568 must be at least divisible by 3.
To check whether a number is divisible by 3 we need to follow the divisibility rule by 3. It says that a number is divisible by 3 when the sum of digits is divisible by 3.
Example: $6561\Rightarrow 6+5+6+1=18$ is divisible by 3.
So, to find whether 2575d568 is divisible by 3 or not we need to check if the sum of digits of the number is divisible by 3 or not.
Therefore, we can write:
$\begin{align}
& \Rightarrow 2+5+7+5+d+5+6+8 \\
& \Rightarrow 38+d \\
\end{align}$
So, we need to put a one-digit value of d that makes $38+d$ divisible by 3.
i.e. d = 1, 4 or 7
If we add d=1, 4 or 7 in $38+d$ we get 39, 42 or 45 which are divisible by 3.
So, for d=1, 4 or 7, the number 2575d568 is divisible by 3. Hence the number would be divisible by 27.
But we need to check if for d=1, 4 or 7 the number 2575d568 is divisible by 29.
Put d=1, 4 or 7 in 2575d568, and check whether the number is divisible by 29. We get to know that for d=7, the number 2575d568 is divisible by 29.
Hence for d=7, the number 2575d568 is divisible by 54 and 87 both.
So, the correct answer is “Option b”.
Note: While checking that if the number 2575d568 is divisible by 3, we get three values of d = 1, 4, and 7. Don’t assume that the number will be divisible by 54 and 87 deliberately. The first check for other factors, i.e. 27 and 29 then confirm if it is divisible or not.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Difference Between Plant Cell and Animal Cell
Which are the Top 10 Largest Countries of the World?
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
How do you graph the function fx 4x class 9 maths CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths