Answer
Verified
91.2k+ views
HINT-Number is exactly divisible by 6 if the number is divisible by 2 and 3.
We have a number of the form$123X43Y$which is divisible by 6.
By the divisibility test of 6 it should be divisible by 2 and 3.
For $123X43Y$to be exactly divisible by 2 the last digit $'y'$should be divisible by 2 so, last digit $'y'$should be multiple of 2 which is$\left\{ {0,2,4,6,8} \right\}$.
Now, to be divisible by 3, the sum of the digits should be divisible by 3.
$\therefore 1 + 2 + 3 + x + 4 + 3 + y = 13 + x + y$
So as we see that $13 + x + y$ is divisible by 3 if the lowest possible value of $x + y$ be 2.
From here the possible cases for $x$ and $y$ is
$
x = 0,y = 2..............\left( 1 \right) \\
x = 2,y = 0...............\left( 2 \right) \\
x = 1,y = 1...............\left( 3 \right) \\
$
So from the following cases the third case is eliminated because $'y'$should be divisible by 2.
So for $x = 0,y = 2$ the number will be$ = 1230432$
And for $x = 2,y = 0$ the number will be$ = 1232430$
Now you have to find out the smallest number in the form $123X43Y$, which is exactly divisible by 6
So, it is clear from the above two numbers that for$x = 0,y = 2$, the number is minimum
So the possible number is$ = 1230432$.
So, this is the required answer.
Note: - In such types of questions the key concept we have to remember is that always remember the divisibility rule of 2, 3 and 6 which is stated above, then according to the divisibility rule find the numbers, then check which number is smaller we will get the required answer.
We have a number of the form$123X43Y$which is divisible by 6.
By the divisibility test of 6 it should be divisible by 2 and 3.
For $123X43Y$to be exactly divisible by 2 the last digit $'y'$should be divisible by 2 so, last digit $'y'$should be multiple of 2 which is$\left\{ {0,2,4,6,8} \right\}$.
Now, to be divisible by 3, the sum of the digits should be divisible by 3.
$\therefore 1 + 2 + 3 + x + 4 + 3 + y = 13 + x + y$
So as we see that $13 + x + y$ is divisible by 3 if the lowest possible value of $x + y$ be 2.
From here the possible cases for $x$ and $y$ is
$
x = 0,y = 2..............\left( 1 \right) \\
x = 2,y = 0...............\left( 2 \right) \\
x = 1,y = 1...............\left( 3 \right) \\
$
So from the following cases the third case is eliminated because $'y'$should be divisible by 2.
So for $x = 0,y = 2$ the number will be$ = 1230432$
And for $x = 2,y = 0$ the number will be$ = 1232430$
Now you have to find out the smallest number in the form $123X43Y$, which is exactly divisible by 6
So, it is clear from the above two numbers that for$x = 0,y = 2$, the number is minimum
So the possible number is$ = 1230432$.
So, this is the required answer.
Note: - In such types of questions the key concept we have to remember is that always remember the divisibility rule of 2, 3 and 6 which is stated above, then according to the divisibility rule find the numbers, then check which number is smaller we will get the required answer.
Recently Updated Pages
Name the scale on which the destructive energy of an class 11 physics JEE_Main
Write an article on the need and importance of sports class 10 english JEE_Main
Choose the exact meaning of the given idiomphrase The class 9 english JEE_Main
Choose the one which best expresses the meaning of class 9 english JEE_Main
What does a hydrometer consist of A A cylindrical stem class 9 physics JEE_Main
A motorcyclist of mass m is to negotiate a curve of class 9 physics JEE_Main
Other Pages
Derive an expression for maximum speed of a car on class 11 physics JEE_Main
If a wire of resistance R is stretched to double of class 12 physics JEE_Main
3 mole of gas X and 2 moles of gas Y enters from the class 11 physics JEE_Main
The vapour pressure of pure A is 10 torr and at the class 12 chemistry JEE_Main
Electric field due to uniformly charged sphere class 12 physics JEE_Main
If the central portion of a convex lens is wrapped class 12 physics JEE_Main