These shortcuts are very useful for solving number theory and algebra questions quickly in competitive exams like SSC, Banking, and Railway exams.

1. Factorization of 1001

The number 1001 has the following factorization:

This identity is very useful for solving divisibility problems.

2. Repeated Block Divisibility Pattern

Example:

This shows that numbers of the form:

are divisible by 1001.

Since:

such numbers are divisible by 7, 11, and 13.

3. Divisibility Rule for 17

Shortcut method:

• Take the last digit of the number
• Multiply it by 5
• Subtract it from the remaining number

Example:

Check if 221 is divisible by 17

Since 17 is divisible by 17, therefore 221 is divisible by 17.

4. Checking if a Number is Prime

Example: Determine whether 177 is prime.

Step 1: Find the nearest square root.

Step 2: Check prime numbers less than or equal to 13.

Test divisibility:

Since 177 is divisible by 3, it is not a prime number.

5. Algebraic Divisibility Identities

The following identities are useful:

1.

for all positive integers m.

2.

when m is even.

3.

when m is odd.

Conclusion

These shortcuts are extremely helpful for solving number theory and algebra questions quickly in competitive exams.

Practicing these identities regularly can significantly improve problem-solving speed.