CDS Mathematics · Number System

Divisibility by Factorisation

Proving what divides a huge expression by factorising it instead of evaluating it — pulling the smallest power out of a sum of like powers, and using the standard a-to-the-n plus-or-minus-b-to-the-n identities.

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Why this matters

Sixteen PYQs and 38% of them HARD — the densest HARD unit in the chapter. But the difficulty is entirely in recognising the shape: every question here is a one-line factorisation followed by reading off a prime factor. The power-sum extraction alone appears six times across the chapter and is the signature CDS pattern in this topic.

Concept 1 of 6

Pulling the smallest power out of a sum of like powers

Intuition

When several terms share a base, factor out the lowest power. What remains is a small integer you can factorise by hand, and that integer carries every odd prime divisor of the whole expression. A thirty-digit number becomes a two-digit one.

Definition

For terms with a common base $a$ ,

a^{m}+a^{m+1}+\cdots+a^{m+k} = a^{m}\left(1+a+a^{2}+\cdots+a^{k}\right).

The bracket is a short geometric sum — compute it as an ordinary number and factorise it.
The power $a^{m}$ contributes only the primes already in $a$ , so any new prime divisor must come from the bracket.
Subtraction works the same way: signs just change the bracket, as in $3^{81}-3^{80}-3^{79} = 3^{79}(9-3-1) = 5\cdot 3^{79}$ .

Power-sum extraction

a^{m}+a^{m+1}+\cdots+a^{m+k}=a^{m}\!\left(1+a+\cdots+a^{k}\right)

Worked example

7^{20}+7^{21}+7^{22}

is divisible by which numbers?

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 1Number SystemEASY

5^{17} + 5^{18} + 5^{19} + 5^{20}

is divisible by

[Q1 · CDS (I) 2018 — Elementary Mathematics · 2018]

A new prime can only come from the bracket

2^{122}+2^{124}+2^{126}+2^{128}+2^{130} = 2^{122}\times 341

, and

341 = 11\times 31

. The answer is 11 — the power of 2 cannot supply an odd factor, so checking the bracket alone is both necessary and sufficient. Students who try to test the whole number for divisibility by 11 get nowhere.

Drill 3 more on pulling the smallest power out of a sum of like powers

Concept 2 of 6

The a-to-the-n minus b-to-the-n identity

Intuition

a^{n}-b^{n}

always has

a-b

as a factor, whatever

n

is. And when

n

is even it also has

a+b

, because you can view it as a difference of squares. So two divisors come for free.

Definition

For all positive integers $n$ ,

(a-b) \mid \left(a^{n}-b^{n}\right).

If $n$ is even, then $a^{n}-b^{n} = (a^{2})^{n/2}-(b^{2})^{n/2}$ is also divisible by $a^{2}-b^{2}$ , hence by $a+b$ .
More generally $a^{m}-b^{m}$ divides $a^{n}-b^{n}$ whenever $m \mid n$ .
Rewrite to expose the shape: $5^{2n}-2^{3n} = 25^{n}-8^{n}$ , so the free factor is $25-8=17$ .

Difference of like powers

a^{n}-b^{n}=(a-b)\left(a^{n-1}+a^{n-2}b+\cdots+b^{n-1}\right)

Worked example

Show that

6^{10}-4^{10}

is divisible by both 2 and 10.

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 2Number SystemMODERATE

The expression

5^{2n} - 2^{3n}

has a factor

[Q50 · CDS (II) 2019 — Elementary Mathematics · 2019]

a plus b needs an EVEN exponent for a difference

97^{30}-14^{30}

is divisible by

97^2-14^2 = 9213 = 3\times37\times83

precisely because 30 is even — which is how 37 and 83 both appear. For an odd exponent only

a-b

is available. Check the parity before claiming

a+b

Drill 3 more on the a-to-the-n minus b-to-the-n identity

Concept 3 of 6

The a-to-the-n plus b-to-the-n identity

Intuition

A sum of like powers factorises only when the exponent is odd, and then

a+b

is a factor. This is the mirror image of the difference rule and the parity condition is reversed, which is the whole trap.

Definition

For odd $n$ ,

(a+b) \mid \left(a^{n}+b^{n}\right).

For even $n$ this fails: $a^{2}+b^{2}$ has no such factorisation over the integers.
The quick way to spot it: add the two bases and see whether the result is one of the options. $41+43 = 84$ ; $67+33 = 100$ ; $327+173 = 500$ .
Both terms being odd also makes the sum even, which is a separate free divisor of 2.

Sum of like powers, odd exponent

a^{n}+b^{n}=(a+b)\left(a^{n-1}-a^{n-2}b+\cdots+b^{n-1}\right) \quad (n \text{ odd})

Worked example

Show that

13^{11}+12^{11}

is divisible by 25.

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 3Number SystemMODERATE

41^{43} + 43^{43}

is divisible by

[Q2 · CDS (I) 2022 — Elementary Mathematics · 2022]

The parity conditions for a sum and a difference are opposite

a-b

divides a difference for every exponent;

a+b

divides a sum only for an odd exponent. Swapping these is the single commonest error in this unit. Write down which you have — sum or difference — then check the exponent's parity against the right rule.

Drill 1 more on the a-to-the-n plus b-to-the-n identity

Concept 4 of 6

Rewriting mixed bases as powers of one number

Intuition

An expression mixing 2, 4 and 8, or 3, 9 and 27, is really a single-base expression in disguise. Convert everything to the smallest base first; the power-sum extraction then applies and the whole thing collapses.

Definition

Replace every composite base by a power of the common base: $4 = 2^2$ , $8 = 2^3$ , $9 = 3^2$ , $27 = 3^3$ , $25 = 5^2$ .

Multiply the exponents: $4^{62} = \left(2^{2}\right)^{62} = 2^{124}$ .
Then sort the terms by exponent and factor out the lowest.
For bases that share a factor without being powers of each other (555 and 777, both multiples of 3 and 37), factor the bases instead and look for a common divisor.

Rebasing

\left(a^{j}\right)^{k}=a^{jk}

Worked example

Simplify

4^{15}+8^{10}

to a single power of 2.

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 4Number SystemHARD

2^{122} + 4^{62} + 8^{42} + 4^{64} + 2^{130}

is divisible by which one of the following integers ?

[Q30 · CDS (II) 2016 — Elementary Mathematics · 2016]

Rebase before you sort, and sort before you factor

2^{122}+4^{62}+8^{42}+4^{64}+2^{130}

looks like five unrelated terms. Rebasing gives

2^{122},2^{124},2^{126},2^{128},2^{130}

— an evenly spaced run — and factoring out

2^{122}

leaves

1+4+16+64+256 = 341 = 11\times31

. Attempting to factor before rebasing gets nowhere.

Drill 1 more on rewriting mixed bases as powers of one number

Concept 5 of 6

The largest number that ALWAYS divides an expression

Intuition

"Always divides" is a claim about every value of the variable — so the answer can be no larger than the value at the smallest admissible input. Compute that one case first; it caps the answer, and usually equals it.

Definition

To find the largest $d$ dividing $f(n)$ for all admissible $n$ :

factor $f(n)$ to get a guaranteed divisor;
evaluate $f$ at the smallest admissible $n$ — the answer must divide that number, which caps it;
confirm the cap is attained.

Watch the domain: "natural number" usually starts at 1, but "whole number" includes 0, and that single difference changes the answer.

The cap from the smallest case

d \mid f(n)\ \forall n \;\Longrightarrow\; d \mid f(n_{\min})

Worked example

What is the largest number that always divides

5^{x}+5^{x+1}

for every natural number

x

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 5Number SystemMODERATE

Which one of the following is the largest divisor of

3^{x} + 3^{x+1} + 3^{x+2}

, if

x

is any natural number ?

[Q10 · CDS (I) 2020 — Elementary Mathematics · 2020]

Whole numbers include zero, and that can break the statement

5\cdot 8^{m}+2^{3m} = 6\cdot 8^{m}

, which is divisible by 48 for every

m \ge 1

— but at

m=0

it equals 6, and 6 is not divisible by 48. Since "whole numbers" include 0, the statement is false. Always test the smallest value the stated domain actually permits.

Drill 1 more on the largest number that always divides an expression

Concept 6 of 6

When a variable divides a polynomial in itself

Intuition

If every term of a polynomial except the constant already has a factor of

x

, then

x

divides the whole thing exactly when

x

divides the constant. So the question collapses to listing the divisors of one small number.

Definition

For a polynomial with integer coefficients,

x \mid \left(a_kx^{k}+\cdots+a_1x+c\right) \iff x \mid c.

Every term containing $x$ is automatically divisible by $x$ ; only the constant can obstruct.
So the count of valid positive $x$ is the number of positive divisors of $c$ .
The same reasoning read backwards handles $\frac{f(m)+n}{m}$ : it is an integer exactly when $m \mid n$ .

Constant-term criterion

x \mid \left(a_kx^{k}+\cdots+a_1x+c\right) \iff x \mid c

Worked example

For how many positive integers

x

x^{2}+3x+12

exactly divisible by

x

Practice this conceptself-check · 4 quick reps

From the bank · past-year question

Example 6Number SystemMODERATE

x^3 + x^2 + 16

is exactly divisible by

x

, where

x

is a positive integer. The number of all such possible values of

x

[Q1 · CDS (II) 2020 — Elementary Mathematics · 2020]

Count the divisors of the constant, not the values you happen to test

For

x^{3}+x^{2}+16

the condition is

x\mid 16

, giving

1,2,4,8,16

— five values. Testing

x=1,2,3,\ldots

by substitution finds the same answers much more slowly and risks stopping early. Reduce to the constant first, then count its divisors.

Drill 1 more on when a variable divides a polynomial in itself

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (6)

Pulling the smallest power out of a sum of like powers
Power-sum extraction
$a^{m}+a^{m+1}+\cdots+a^{m+k}=a^{m}\!\left(1+a+\cdots+a^{k}\right)$
The a-to-the-n minus b-to-the-n identity
Difference of like powers
$a^{n}-b^{n}=(a-b)\left(a^{n-1}+a^{n-2}b+\cdots+b^{n-1}\right)$
The a-to-the-n plus b-to-the-n identity
Sum of like powers, odd exponent
$a^{n}+b^{n}=(a+b)\left(a^{n-1}-a^{n-2}b+\cdots+b^{n-1}\right) \quad (n \text{ odd})$
Rewriting mixed bases as powers of one number
Rebasing
$\left(a^{j}\right)^{k}=a^{jk}$
The largest number that ALWAYS divides an expression
The cap from the smallest case
$d \mid f(n)\ \forall n \;\Longrightarrow\; d \mid f(n_{\min})$
When a variable divides a polynomial in itself
Constant-term criterion
$x \mid \left(a_kx^{k}+\cdots+a_1x+c\right) \iff x \mid c$

Watch out for (6)

A new prime can only come from the bracket
→ Pulling the smallest power out of a sum of like powers
a plus b needs an EVEN exponent for a difference
→ The a-to-the-n minus b-to-the-n identity
The parity conditions for a sum and a difference are opposite
→ The a-to-the-n plus b-to-the-n identity
Rebase before you sort, and sort before you factor
→ Rewriting mixed bases as powers of one number
Whole numbers include zero, and that can break the statement
→ The largest number that ALWAYS divides an expression
Count the divisors of the constant, not the values you happen to test
→ When a variable divides a polynomial in itself

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