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Category: SSC

  • Number Theory and Algebra Shortcut Notes

    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:

    1001=7×11×131001 = 7 \times 11 \times 13

    This identity is very useful for solving divisibility problems.

    2. Repeated Block Divisibility Pattern

    Example:

    1001×234=2342341001 \times 234 = 234234
    1001=7×11×131001 = 7 \times 11 \times 13

    This shows that numbers of the form:

    abcabcabcabc

    are divisible by 1001.

    Since:

    1001=7×11×131001 = 7 \times 11 \times 13

    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

    22−(1×5)=1722 – (1 \times 5) = 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.

    177≈13\sqrt{177} \approx 13

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

    Test divisibility:

    177÷3=59177 \div 3 = 59

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

    5. Algebraic Divisibility Identities

    The following identities are useful:

    1.

    xm−am is divisible by (x−a)x^m – a^m \text{ is divisible by } (x-a)

    for all positive integers m.

    2.

    xm−am is divisible by (x+a)x^m – a^m \text{ is divisible by } (x+a)

    when m is even.

    3.

    xm+am is divisible by (x+a)x^m + a^m \text{ is divisible by } (x+a)

    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.

  • Algebraic Identities for x + 1/x – Important Formulas for Exams

    In many competitive exams like SSC, Banking, and Railway exams, expressions involving x + 1/x frequently appear. These identities help simplify such expressions quickly.

    1. Basic Identity

    If:

    x + 1/x = a

    Then the following identities are useful:

    x² + 1/x² = a² − 2

    x³ + 1/x³ = a³ − 3a

    These formulas are commonly used in algebra simplification problems.

    2. When x − 1/x = a

    If:

    x − 1/x = a

    Then:

    x² + 1/x² = a² + 2

    x³ − 1/x³ = a³ + 3a

    These identities are also useful in competitive exam problems.

    3. Fourth Power Identity

    The following factorization identity is useful:

    a⁴ + a²b² + b⁴ = (a² + ab + b²)(a² − ab + b²)

    4. Fifth Power Relation

    For expressions involving x + 1/x:

    x⁵ + 1/x⁵ = (x² + 1/x²)(x³ + 1/x³) − (x + 1/x)

    5. Special Values

    ValueResult
    x + 1/x = 2x = 1
    x + 1/x = −2x = −1
    x + 1/x = 1x³ = −1
    x + 1/x = −1x³ = 1

    6. General Identity

    If:

    ax + b/x = k

    Then:

    a²x² + b²/x² = k² − 2ab

    a³x³ + b³/x³ = k³ − 3abk

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