 # Number System Problems with Solutions

Try to understand these problems on Numbers:
1. Example 1: The product of 4 consecutive even numbers is always divisible by A. What is A?

The smallest possible 4 consecutive even numbers are: 2, 4, 6 and 8. The product of these four numbers is 2 × 4 × 6 × 8 = 384. So, any 4 consecutive numbers will always be divisible by 384.
2. Which least digit should replace the \$ in the number 23\$788, so that it becomes a multiple of 3.

As you know that if the sum of all the digits is divisible by 3, then the number is divisible by 3. Now sum of the given digits is 2 + 3 +\$+ 7 + 8 + 8 = 28+ \$. Now think the next multiple of 3 after 28 i.e. 30. This means you add 2 in this. The value of \$ is 2.
3. Find the greatest three-digit number which is a multiple of 13.

Greatest three digit number is 999. When we divide 999 by 13, then 11 is the remainder. So, 999 – 11 = 988 is the answer.
4. Find the smallest 6-digit number, which is a multiple of 18.

To solve such question take the smallest six-digit number, which is 100000. Divide this number by 18 and get the remainder as 10. Here if you subtract 10 from the number, no doubt you will get a multiple of 18. But because you have already taken the smallest six-digit number, if you subtract anything from it, you will get a five-digit number. Think it otherwise, that instead of subtracting you add something.
Now what should be added to 10(the remainder) so that it becomes a multiple of 18? i.e. 18 – 10 = 8 ⇒ 8 should be added in the number i.e. 100000 + 8 = 100008 is the answer.
5. Find the sum of odd natural numbers up to 100.

Here we have 1 + 3 + 5 + ...... + 99 = (50 × 100/2) = 2500
6. What is the value of P, where P = 13 + 23 +......203 ?

You have to find the sum of first 20 perfect cubes. The formula of sum of cubes of 1st N Natural Numbers is to be applied. ∑203 =((20 × 21 )/ 2)2 = 44100.
7. How many total squares are there in 4 x 4 square? The formula for number of squares in n x n square is ∑n2 and ∑n2 =(n(n+1)(2n+1)/6) Here n is 4. Putting n = 4, we get 30 as answer.
1. Find the smallest 8-digit number which is a multiple of 9

Smallest eight-digit number is 10000000, when we divide 10000000 by 9, then 1 is the remainder. So, 10000000 – 1 + 9 = 10000008 is the answer. (1 is subtracted to find the multiple of 9, as 1 is the remainder, but then 9 is added to get the smallest such eight-digit number, otherwise you were having a seven-digit number).
2. The sum of two numbers is 42. What could be the maximum possible product of those 2 numbers?

When the sum of two numbers is constant, then product of those two numbers is maximum, when those two numbers are as close to each other. As 21 + 21 = 42, so answer will be 21 × 21 = 441.
3. There are two numbers, such that one number is 6 more than the other number. If the total of two numbers is 18, find the product of those numbers.