Question

A boy multiplied 987 by a certain number and obtained 559981 as his answer. If in the answer both 9 are wrong and the other digits are correct, then the correct answer would be:
(a) 553681
(b) 555181
(c) 555681
(d) 556581

Answer 1

Hint: Write the prime factorization of 987. Use the fact that if a number is divisible by 987, it will be divisible by all the prime factors of 987. Check the divisibility of all the options from the prime factorization of 987 to get the number which is divisible by 987.

Step-by-step answer:
We have to find the number divisible by 987. To do so, we will first write the prime factorization of 987.
To write the prime factorization of any number, start by dividing the number by the first prime number, which is 2 and then continue to divide by 2 until you get a number which is not divisible by 2 (which means that you get a decimal or remainder on dividing the number by 2). Then start dividing the number by the next prime number which is 3. Continue dividing the number by 3 until you get a number which is not divisible by 3. Thus, continuing the process, keep dividing the numbers by series of prime numbers \[5,7,...\]until the only numbers left are prime numbers. Write the given number as a product of all the prime numbers (considering the fact to count each prime number as many times as it divides the given number) to get the prime factorization of the given number.
Thus, the prime factorization of 987 is $987=3\times 7\times 47$.
We know that if a number is divisible by 987, it will be divisible by all the prime factors of 987.
We will now check the divisibility of all the numbers by 3. We know that a number is divisible by 3 if the sum of its digits is divisible by 3.
We will now check each of the options.
We will firstly consider 553681. The sum of its digits is $5+5+3+6+8+1=28$. As 28 is not divisible by 3, the number 553681 is not divisible by 987.
We will now consider the number 555181. The sum of its digits is $5+5+5+1+8+1=25$. As 25 is not divisible by 3, the number 555181 is not divisible by 987.
We will now consider the number 555681. The sum of its digits is $5+5+5+6+8+1=30$. As 30 is divisible by 3, we will check if 555681 is divisible by 7 and 47. Dividing the number 555681 by 7, we have $\dfrac{555681}{7}=79383$. Dividing the number 555681 by 47, we have $\dfrac{555681}{47}=11823$.
As 555681 is divisible by all the prime factors of 987, it is divisible by 987 as well.
We will now consider the number 556581. The sum of its digits is $5+5+6+5+8+1=30$. As 30 is divisible by 3, we will check if 556581 is divisible by 7 and 47. Dividing the number by 7, we have $\dfrac{556581}{7}=79511.57$. As 556581 is not divisible by 7, it is not divisible by 987 as well.
Hence, the correct answer is 555681, which is option (c).

Note: We can also solve this question by dividing each of the given numbers by 987 and checking if they are divisible or not. However, as 987 is a large number, one must know the divisibility rule to check if the numbers are divisible by 3.