Theory on Number System

Unit digit theorem is just extension of Remainder Theorem but the change is that you always need to divide given expression by 10 to find its unit digit. On the logic of remainder theorem, the unit's digit of an expression will be taken by getting the remainder when the expression is divided by 10.

Remainder theorem used to find remainder of large expressions or expression having large power.

Say, a number 37 is divided by 7 then what would be the remainder? Yes, 2 is the actual remainder which is also a positive remainder. It also gives a negative remainder. The negative remainder would be -5. See, the positive remainder is 2 which are just more than 35 and -5 is just less than 42. Notice, 35 and 42 is divisible by 7.

There is two types of remainder theorem.Positive Remainder Theorem and Negative Remainder Theorem. Here we will understand the basics of positive remainder theorem.

• When a perfect square is written as a product of its prime factors each prime factors will appear an even number of times. E.g.
  7016 = 2 × 2 × 2 × 2 × 21 × 21.

Squaring is multiplying the numbers by itself.

Counting is a process to count numbers between two given numbers.

A number is said to be factor or sub-multiples of another number when it is exactly divides the other number i.e. the remainder obtained is zero.

Where [x], denotes the greatest integer less than or equal to x. Actually, we are dividing the given number inside the factorial by 5, 25, 125 and so on. We use this process till we don't get division result less than 1.

Let a given expression is 13 × 15 × 22 × 125 × 44 × 35 × 11. You can see given expression is a products of numbers.

Any composite number can be written as the products of its prime factors, this product is called Standard form of the number.

A number (dividend) is said to be divisible by another number (called divisor) when quotient is a natural number and remainder is zero. In other words, we can say that when a number (dividend) is divisible by another number (divisor), the dividend can be expressed as multiple of divisor.