Key facts

  • If N = p^a q^b r^c, the number of positive factors is (a + 1)(b + 1)(c + 1).
  • The sum of factors is found by multiplying prime-power sums such as (1 + p + p^2 + ... + p^a).
  • For 120 = 2^3 x 3 x 5, the number of positive factors is 16 and the sum of all positive factors is 360.
  • Odd factors are counted or summed by ignoring the power of 2 in the prime factorisation.
  • The difference of squares identity is a^2 - b^2 = (a - b)(a + b); a sum of squares is not treated the same at this level.

Key Points at a Glance

  1. 1

    A factor divides a number or expression exactly; a multiple is obtained by multiplying by an integer.

  2. 2

    Prime factorisation converts divisor questions into exponent choices, making count and sum formulas reliable.

  3. 3

    If N = p^a q^b r^c, the number of positive factors is (a + 1)(b + 1)(c + 1).

  4. 4

    The sum of factors is found by multiplying prime-power sums such as (1 + p + p^2 + ... + p^a).

  5. 5

    For 120 = 2^3 x 3 x 5, the number of positive factors is 16 and the sum of all positive factors is 360.

  6. 6

    Odd factors are counted or summed by ignoring the power of 2 in the prime factorisation.

  7. 7

    In algebraic factorisation, first take the common factor before applying grouping or identities.

  8. 8

    Grouping works when paired terms produce the same bracket after taking local common factors.

  9. 9

    The difference of squares identity is a^2 - b^2 = (a - b)(a + b); a sum of squares is not treated the same at this level.

  10. 10

    A perfect-square trinomial must have square end terms and a middle term equal to twice the product of their roots.

  11. 11

    Quadratic trinomials are factorised by product-sum thinking or by splitting the middle term when the leading coefficient is not 1.

  12. 12

    By the basic factor theorem, if p(a) = 0, then x - a is a factor of p(x).

  13. 13

    In simplification, cancel only common factors, never isolated terms inside addition or subtraction.

  14. 14

    For factor-based equations, move all terms to one side, factorise fully and then apply the zero-product rule.

How should LDC candidates treat factors and polynomial factorisation in the exam?

LDC candidates should treat factors and polynomial factorisation as one speed-and-accuracy unit: numerical divisor work, common-factor extraction, identities and quadratic splitting all test the same idea of exact division. This topic joins two surfaces that look different in the book but behave similarly in the exam: numerical factors and algebraic factorisation. In the official LDC mathematics syllabus, factors are named separately from polynomial factorisation, so preparation should not stop at arithmetic divisibility. The recent CET Senior Secondary paper signal also supports this balance: one direct question asked for the sum of all factors of 120, while polynomial factorisation remains a syllabus-driven area where quick algebraic recognition can still decide a routine MCQ. Treat the topic as a speed-and-accuracy unit, not as a proof-heavy algebra chapter. According to the Rajasthan Staff Selection Board CET Senior Secondary 2024 syllabus, the paper contains 150 questions.

A factor of a number is a number that divides it exactly. For example, 6 is a factor of 42 because 42 divided by 6 gives 7 with no remainder. A multiple moves in the opposite direction: 42 is a multiple of 6 because it can be written as 6 times 7. In arithmetic questions, the trap is usually language. "Factors of 120" asks for divisors of 120; "multiples of 120" asks for numbers like 120, 240, 360 and so on. Factors are limited for a fixed positive integer; positive multiples are unlimited.

In algebra, a factor is an expression that divides another expression exactly. If x squared minus 9 is written as (x - 3)(x + 3), then x - 3 and x + 3 are factors of that polynomial. The same idea of exact division remains, but numbers are replaced by expressions. A common factor is the part shared by every term: in 6x squared plus 9x, the common factor is 3x, so the expression becomes 3x(2x + 3). This is the algebraic version of taking common prime parts out of numbers.

The central exam skill is to choose the method before doing long work. If the question is numerical, first look for prime factorisation and divisor formulas. If the question is algebraic and every term shares something, first take the common factor. If terms are four in number, grouping may be faster. If the expression matches an identity such as a squared minus b squared, use the identity immediately. If a quadratic trinomial has the form ax squared plus bx plus c, use product-sum thinking to split the middle term.

A useful mental checkpoint is: a correct factorisation must multiply back to the original expression. Many wrong options in MCQs are designed around sign errors. For example, (x - 5)(x - 2) gives x squared - 7x + 10, not x squared + 7x + 10. Similarly, (a - b) squared equals a squared - 2ab + b squared, not a squared - b squared. The minus sign does not simply pass through the square.

For LDC-level numeracy, build a small sequence for every question. First identify whether it is number-based or expression-based. Second, write the smallest necessary factor form. Third, apply the formula or identity. Fourth, check the answer by substitution, multiplication, or divisibility. This four-step discipline keeps the topic fast without making it mechanical. It also prevents the common mistake of memorising formulas while missing the exact wording of the question.