Question

If $2576a456b$ is divisible by $15$, then

• A. $a$ may take the value $5$.
• B. $b$ may take the value $0$.
• C. $a$ may take the value $4$.
• D. $a$ may take the value $6$.

Answer 1

Answer: (A), (B), (C)

The correct options are
A $a$ may take the value $5$.
B $b$ may take the value $0$.
C $a$ may take the value $4$.

The given number is $2576a456b$

To be divisible by $15$, the number has to divisible by both $3$ and $5$.

As per divisibility rule for $5$, the last digit should be $0$ or $5$

so $b$ can be $0$ or $5$

So option $B$ is correct.

Now we take both cases of value of $b.$

Case - 1: when $b=0$

Now the given number becomes $2576a4560$

For a number to be divisible by $3,$ the sum of its digits should be divisible by $3.$

Thus the sum of digits of this number is $35+a$

For $35+a$ to be divisible by $3,$ a can be anything from $1,4,7,\mathrm{10....}$

So option $C$ is correct.

Now Case-2: when $b=5$

Now the given number becomes $2576a4565$

Thus the sum of digits of this number is $40+a$

For $40+a$ to be divisible by $3,$ a can be anything from $2,5,8,\mathrm{11....}$

So option $A$ is correct.