Number System Basics – Logical Reasoning Study Notes

Understanding Fundamental Number Classifications

At the heart of any numerical analysis lies the distinction between various types of numbers. A Prime Number is defined as a natural number greater than 1 that possesses exactly two distinct factors: 1 and itself. Understanding prime numbers is crucial because they act as the “building blocks” of all integers through prime factorization.

In contrast, Composite Numbers are natural numbers that have more than two factors. It is a common point of confusion, but remember that the number 1 is neither prime nor composite. The number 2 holds a unique position in mathematics as the only even prime number; all other prime numbers are odd.

Note: Any prime number greater than 3 can be expressed in the form of 6n ± 1, where ‘n’ is a natural number. While this is a helpful heuristic for identification, it is not a definitive test for primality, as not every number in this form is necessarily prime.

The Prime Testing Algorithm (Root Method)

When you encounter large numbers in an exam, you don’t need to check every divisor. The Root Algorithm provides a systematic way to test for primality. First, calculate the approximate square root of the target number ‘n’. If the square root is not a perfect integer, round it up to the next natural number.

Once you have this limit, identify all prime numbers smaller than that value. Divide your target number by each of these primes sequentially. If the number is divisible by any of these primes, it is composite. If no division results in a remainder of zero, the number is confirmed as prime.

• Example: To check if 241 is prime, find its root (~15.5, round to 16).
• Test divisors: 2, 3, 5, 7, 11, and 13.
• Since 241 is not divisible by any of these, it is confirmed as a prime number.

Summation Formulas for Competitive Exams

UPSC and SSC exams often test your ability to handle series and sequences. Memorizing specific summation formulas can save significant time during the paper. These formulas allow you to calculate the sum of natural numbers, squares, and cubes without manual addition.

1. Sum of first ‘n’ natural numbers: S = n(n + 1) / 2
2. Sum of squares of first ‘n’ natural numbers: S = n(n + 1)(2n + 1) / 6
3. Sum of cubes of first ‘n’ natural numbers: S = [n(n + 1) / 2]²
4. Sum of first ‘n’ odd natural numbers: S = n²

Algebraic Properties of Digits

Logical reasoning questions often involve the rearrangement of digits in two-digit or three-digit numbers. A two-digit number can be represented as 10x + y, where ‘x’ is the tens digit and ‘y’ is the units digit. When you reverse these digits, the new number becomes 10y + x.

The difference between these two numbers is always a multiple of 9, specifically 9|x – y|. Similarly, the sum of a two-digit number and its reverse is always a multiple of 11, expressed as 11(x + y). These properties are frequently used in “missing digit” puzzles and cryptarithmetic problems.

Important Facts and Formulas

Previous Year Question Hints

• Sum of Primes: If the sum of three prime numbers is an even number (e.g., 100), one of the numbers must be 2. This is because the sum of three odd numbers is always odd, and the only way to get an even sum from three primes is to include the only even prime, 2.
• Sequence Patterns: When asked to find the last term of a row in a triangular arrangement (like 1; 2,3; 4,5,6), recognize that the last term of the nth row is the sum of the first n natural numbers: n(n+1)/2.

Quick Revision Summary

• 1 is neither prime nor composite.
• 2 is the only even prime number.
• There are 25 prime numbers up to 100 and 46 up to 200.
• Co-prime numbers have an HCF of 1.
• The Root Algorithm is the most efficient way to test for large primes.
• Sum of first ‘n’ odd numbers is always n².
• The difference between a 2-digit number and its reverse is always a multiple of 9.
• The sum of a 2-digit number and its reverse is always a multiple of 11.