Question

If 2576a456b is divisible by 15, then which of the following options hold.
This question has multiple correct options
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

Hint: Divisibility rule:
 To be divisible by 5 a number must end in 0 or 5.
To be divisible by 3 the rule is to add up all the digits of the number if it is a multiple of 3 then the original number is also divisible by 3.
To be divisible by 15 a number has to be divisible by 3 and 5.
So if a number satisfies the divisibility rule of 5 and 3 then it is divisible by 15.

Complete step-by-step answer:
It is given that 2576a456b is divisible by 15.
We know that, if a number satisfies the divisibility rule of 5 and 3 then it is divisible by 15.
Since 2576a456b is divisible by 15, then by divisibility rule it is divisible by 5 and 3.
Now, 2576a456b to be divisible by 5 then it must end in 0 or 5.
Then b must be 0 or 5.
Thus we can say that the value of b may be 0.
So (B) is correct.

Again, To be divisible by 3 the rule is to add up all the digits of the number if it is a multiple of 3 then the original number is also divisible by 3.
Applying this rule for 2576a456b we get,
\[2 + 5 + 7 + 6 + a + 4 + 5 + 6 + b = 35 + a + b\]
Since 2576a456b is divisible by 3, then \[35 + a + b\]must be multiple of 3.
We already get that b is either 0 or 5.

If \[b = 0\] then the sum of the digits becomes \[35 + a\]and it must be multiple of 3.
If the value of a is 5 then \[35 + a\]becomes 40 which is not divisible by 3.
So, (A) is not correct.

If the value of a is 4 then \[35 + a\]becomes 39 which is divisible by 3
Then the value of a can be 4.
So, (C) is correct.

If the value of a is 6 then \[35 + a\]becomes 41 which is not divisible by 3.
So, (D) is not correct.

Hence, (B) and (C) are the correct options.

Note:
Divisibility rule plays a major role in finding the values of a and b. Any number divisibility rules can be obtained by taking the reference from their prime factors. For example in the problem we found prime factors of 15 are 3 and 5. So, to be divisible by 15 it must be divided by both 3 and 5.