Aptitude · Foundations

91 is not prime, and knowing why saves you a minute a paper

Q: Is 2 prime? And are there other even primes?

Yes, 2 is prime — its only factors are 1 and 2. It is the only even prime, because any other even number has 2 as a factor in addition to 1 and itself, giving it at least three factors.

Before percentages, ratios or interest, every quantitative paper assumes you can classify a number on sight and predict whether a result will be odd or even. Both skills eliminate wrong options without solving anything.

Pick a number and watch the primality test stop early →

Parity is the cheapest tool in aptitude. Knowing only whether an answer must be odd or even often removes half the options before you have done any real work.

01 The idea

Three classifications and one habit

An even number is one that 2 divides exactly: 0, 2, 4, 6, 8 and everything ending in those digits. An odd number is one that 2 does not divide. Nothing about that depends on the size of the number, only on its last digit, which is why parity is so cheap to check.

A prime number is one with exactly two factors — 1 and itself. So 2, 3, 5, 7, 11 are prime. Two consequences catch people out. One, 1 is not prime, because it has only one factor. Two, 2 is prime, and it is the only even prime, because every other even number has 2 as a third factor.

The trap is assuming an odd number that looks awkward is prime. 91 looks prime. It is 7 × 13. So does 51, which is 3 × 17, and 87, which is 3 × 29. Placement papers use exactly these numbers, because a student who guesses gets them wrong.

The habit worth building is the opposite of solving: before you calculate an answer, ask what kind of answer it must be. If a question's answer must be even and three of four options are odd, you are finished without arithmetic. That is what the parity rules in section 03 are for.

A prime has exactly two factors. 1 is not prime, 2 is the only even prime, and an odd number that looks awkward is usually composite — check it rather than assuming.

Even and oddEven means 2 divides it exactly; odd means it does not. Decided entirely by the last digit, so it is free to check no matter how large the number is.

Prime and compositeA prime has exactly two factors, 1 and itself. A composite has more. 1 is neither — it has one factor, so it fails the definition of prime.

ParityWhether a number is odd or even, treated as a property you can track through a calculation. Addition and multiplication both have fixed parity rules, which is what makes it useful.

02 Worked example

Is 91 prime? Four divisions settle it

The point of this example is not the answer but the stopping rule. Most students test far too many divisors, or too few and guess. Determine whether 91 is prime.

Check the parity91 ends in 1, so it is odd. That rules out 2 as a factor — and rules out nothing else.91 is odd ⇒ 2 is not a factor

Find how far you must testYou only need divisors up to the square root of 91. Ten squared is 100, which overshoots, so the limit is 9.√91 ≈ 9.54 ⇒ test primes up to 9: that is 3, 5 and 7

Try 3, using the digit-sum ruleThe digits of 91 add to 10, and 3 does not divide 10. So 3 is not a factor, and no division was needed.9 + 1 = 10, and 10 is not a multiple of 3 ⇒ 3 is out

Try 5A multiple of 5 ends in 0 or 5. 91 ends in 1.91 does not end in 0 or 5 ⇒ 5 is out

Try 7 — and there it isSeven divides 91 exactly, giving 13. The moment a single factor appears, the number is composite and you stop.91 / 7 = 13 exactly ⇒ 91 = 7 × 13, composite

Why stop at the square root? Because if 91 had a factor above 9.54, that factor would have to multiply by something below 9.54 to make 91 — and you would already have found the smaller one. Factors come in pairs straddling the root. That is the reason, and it turns primality from an endless check into three or four divisions.

03 The method

The parity rules, and why each one is true

Most students memorise this table and forget it. Each row has a one-line reason, and the reasons are easier to keep than the rows.

Write an even number as 2k and an odd number as 2m + 1. Then every rule below is one line of algebra. Odd + Odd = (2m+1) + (2n+1) = 2(m+n+1), which has a factor of 2 — so it is even.

For addition and subtraction, only the count of odd numbers matters. Add up any list: the total is odd exactly when the number of odd terms in it is odd. For multiplication, one single even factor makes the whole product even — so a product is odd only when every factor is odd.

Operation	Result	Why
Odd ± Odd	Even	(2m+1) ± (2n+1) loses the two 1s, leaving a multiple of 2
Odd ± Even	Odd	The lone +1 survives, so the total cannot be a multiple of 2
Even ± Even	Even	2m ± 2n = 2(m ± n), still a multiple of 2
Odd × Odd	Odd	(2m+1)(2n+1) = 2(…) + 1 — the +1 × +1 is left over
Odd × Even	Even	The even factor contributes a 2 that nothing removes
Even × Even	Even	Two factors of 2, so divisible by 4 as well
Division	No rule	The result need not even be a whole number, so parity does not apply

05 Cheat sheet

Foundations on one page

The first three rows are the classifications. The rest is what you use them for, which is eliminating options rather than computing answers.

Fact	Rule	Watch out for
Even	`last digit in 0 2 4 6 8`	Size is irrelevant — only the last digit
Prime	`exactly 2 factors`	1 is not prime
Only even prime	`2`	Every other even number has 2 as a third factor
Primality test limit	`test primes up to √n`	Factors pair around the root, so above it is wasted work
Sum of a list	`odd iff an odd count of odds`	Even terms never change parity
Product of a list	`odd iff every factor is odd`	One even factor makes it even
Division	`no parity rule`	The answer may not be a whole number at all

The false primes worth memorising91 = 7×13, 51 = 3×17, 87 = 3×29, 143 = 11×13, 133 = 7×19. Each looks prime and each appears in placement papers for that reason.

Parity eliminates before it calculatesIf the answer must be even, every odd option is gone. Ask what kind of answer the question forces before you start working it out.

Only test prime divisorsA composite divisor would have been caught by its own prime factors already, so testing 4 after 2, or 9 after 3, is duplicated work.

06 Where & why

Where this actually earns marks

None of this is a question type on its own. It is the layer underneath the question types, and it shows up as speed rather than as a topic.

TCS NQT · number-system questions

“How many primes between…”

Directly tested, and the square-root stopping rule is the difference between finishing and running out of time.

Any paper · option elimination

Parity of the answer

In ratio, age and coin questions the answer must often be a whole number of a particular parity. Two options usually die immediately.

HCF, LCM and simplification

Factorising quickly

Every ratio and fraction question needs numbers broken into factors. Recognising 91 as 7 × 13 on sight is what makes cancelling fast.

Divisibility questions

The rules in the next lesson

The families of divisibility rules are built on these classifications, and they are the fastest way to find a factor without dividing.

The measure of whether this lesson has landed is not whether you can recite the parity table. It is whether, seeing 91 in a question, you factor it instead of assuming it is prime.

07 Interview questions

What gets asked about numbers

Nine questions, from the definitions to the reasoning that makes the primality test fast.

Is 1 a prime number?

No. A prime has exactly two distinct factors, and 1 has only one — itself. It is not composite either, since composite means more than two factors. So 1 sits outside both categories, which is also why it is excluded from unique prime factorisation.

Is 2 prime? And are there other even primes?

Yes, 2 is prime — its only factors are 1 and 2. It is the only even prime, because any other even number has 2 as a factor in addition to 1 and itself, giving it at least three factors.

How far do you have to test to decide whether a number is prime?

Up to the square root, and only prime divisors. If n = a × b then at least one of a and b is at most √n, so a factor larger than the root always pairs with a smaller one you have already tried. For 91 that means testing only 3, 5 and 7.

Why only prime divisors?

Because a composite divisor is built from primes you have already tested. If 4 divided the number then 2 would have, and if 9 divided it then 3 would have. Testing composites duplicates work you have done.

Is 91 prime?

No, 91 = 7 × 13. It is the standard trap because it is odd, it is not obviously divisible by 3 or 5, and it sits just past where casual checking stops. The same applies to 51 = 3 × 17, 87 = 3 × 29 and 143 = 11 × 13.

Why is odd plus odd always even?

Write the two odds as 2m + 1 and 2n + 1. Adding gives 2m + 2n + 2, which is 2(m + n + 1) — a multiple of 2, so even. The two spare 1s pair up into another 2, which is the whole reason.

What is the parity of a product, in general?

A product is odd only if every single factor is odd. One even factor contributes a 2 that nothing else can remove, so the whole product becomes even. That is why checking a long product for parity means scanning for a single even term rather than tracking everything.

Why is there no odd-even rule for division?

Hard

Find the smallest whole number greater than 1000 that is odd, is not prime, and has no prime factor smaller than 10. Show that every odd number between 1001 and your answer fails at least one of the three conditions, and say which.

Follow-up

The conditions pull against each other: ruling out 2, 3, 5 and 7 as factors while still insisting the number be composite leaves very little room. Note that 1001 looks like the obvious answer and is not — it is 7 × 11 × 13, and that 7 disqualifies it.

Show the hint

Walk up in odd steps from 1001. For each one, either it has a factor below 10, or it is prime — until one is neither.