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Number Theory

Number Theory

MODULES

Basics of Numbers
Types of Numbers
Fractions
Arithmetic Operations
Other Numerical Operations
Algebraic Expansion
Prime Numbers
Counting Integers
Past Questions

CONCEPTS & CHEATSHEET

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Number Theory 1
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PRACTICE

Number Theory : Level 1
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ALL MODULES

CAT 2025 Lesson : Number Theory - Prime Numbers

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5. Prime and Composite numbers

5.1 Prime numbers

Prime numbers are those natural numbers that can be perfectly divided by exactly two positive integers, 111 and the number itself. In other words, prime numbers have exactly two factors. The increasing sequence of prime numbers is 222, 333, 555, 777, 111111, 131313, 171717, 19...19 ...19...
The number 111 is neither prime nor composite.

Some key points to note here are:

1) No pattern has been established for the occurrence of prime numbers.
2) All prime numbers are odd numbers, with the exception of 222.
3) All prime numbers except 222 and 555 end with the digits 111, 333, 777 or 999.
4) For ease in exams, note that there are 151515 prime numbers less than 505050 and 252525 prime numbers less than 100100100.
5) All prime numbers, other than 222 and 333, are in the form (6k+1)(6k + 1)(6k+1) or (6k−1)(6k - 1)(6k−1), where k is a positive integer.

5.2 Composite numbers

Composite numbers are natural numbers that have more than two factors. These are numbers formed when two or more prime numbers are multiplied. 4=2×24=2 \times 24=2×2, 105=3×5×7105=3 \times 5 \times 7105=3×5×7. Composite numbers can be odd or even (eg., 444, 999)

5.3 Test for Prime

To check whether a number is prime or not:

Step 1: Calculate the square root of the number.
Step 2: If the square root is an integer, then the number is not prime. If not,
Step 3: Determine all prime numbers less than or equal to the square root
Step 4: Check if any of these prime numbers perfectly divide the number
Step 5: If none of them divide the number, then the number is prime, else the number is composite.

Example 15

Which of 143143143, 323323323 and 331331331 are prime numbers?

Solution

143<144=12\sqrt{143} \lt \sqrt{144} = 12143​<144​=12
Prime numbers less than 121212 are 222, 333, 555, 777 and 111111.
143\bm{143}143 is divisible by 111111 and ∴\therefore∴ not prime.

323<324=18\sqrt{323} \lt \sqrt{324} = 18323​<324​=18
Prime numbers less than 181818 are 222, 333, 555, 777, 111111, 131313 and 171717.
323\bm{323}323 is divisible by 171717 and ∴\therefore∴ not prime.

331<361=19\sqrt{331} \lt \sqrt{361} = 19331​<361​=19
Prime numbers less than 191919 are 222, 333, 555, 777, 111111, 131313 and 171717.
331\bm{331}331 is not divisible by any of these and ∴\therefore∴ is prime.

Answer: 331331331 only


5.4 Co-prime Numbers

When two numbers do not have any common factors except for 111, they are said to be co-primes as they are prime relative to each other. For instance, 444 and 999, though composite numbers, are co-prime numbers as they do not have any common factors aside from 111.

Example 16

Which of the following options have a pair of numbers that are co-prime?

(1) 121212 and 212121           (2) 181818 and 494949           (3) 242424 and 383838           (4) 373737 and 111111111

Solution

Co-primes do not have any common factor.

Option 1: 333 is a common factor.
Option 2: No common factor. ∴\therefore∴ Co-prime

Option 3: 222 is a common factor.

Option 4: 373737 is a common factor.

Answer: (2) 181818 and 494949

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