Factors and multiples — all factors of a number, factor pairs

1.1.7 · Maths › Arithmetic & Number Systems

1. Core definitions

WHAT har number ke factors ke baare mein sach hai:

$1$ har number ka factor hota hai ( $n = 1 \times n$ ).
$n$ khud apna factor hota hai ( $n = n \times 1$ ).
Har factor $a$ , $1 \leq a \leq n$ satisfy karta hai.

2. "Find all factors" method scratch se derive karna

HOW karte hain hum har factor find, bina koi miss kiye? Aao trick derive karein.

Maano $a$ , $n$ ka factor hai, to $n = a \times b$ . Tab $b = n / a$ automatically bhi ek factor hai — yeh dono pairs mein aate hain. Ab poochho: ek pair ke chhote member ki maximum size kya ho sakti hai?

Agar dono $a$ aur $b$ , $n$ se bade hote, to $a \times b > n \times n = n,$ jo $a \times b = n$ se contradict karta hai. Isliye har factor pair ka kam se kam ek member $\leq n$ hota hai.

Figure — Factors and multiples — all factors of a number, factor pairs

3. Worked examples

4. Common mistakes ko steel-man karna

5. Active recall

Recall Quick self-test (answers chhupa lo!)

Ek prime ke kitne factors hote hain? → exactly 2.
Hum sirf $n$ tak kyon search karte hain? → har factor pair ka ek member $\leq n$ hota hai.
Perfect squares mein odd number of factors kyon hote hain? → pair $(n, n)$ ek factor hai jo khud se pair karta hai.
$30$ ke factor pairs? → $(1, 30), (2, 15), (3, 10), (5, 6)$ .

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho tumhare paas $24$ identical square tiles hain aur tum ek perfect rectangle floor banana chahte ho. Tum $1 \times 24$ , $2 \times 12$ , $3 \times 8$ , ya $4 \times 6$ bana sakte ho — har woh rectangle jo saare tiles ko neatly use kare. Un rectangles ki side lengths hi 24 ke factors hain! Inhe jaldi dhundhne ke liye, sirf chhoti sides ko us point tak try karo jahan rectangle square ban jaye — uske baad tum wohi rectangles flip kar rahe ho jo already mil chuke hain. Agar shape ek perfect square ban sakti hai (jaise $36 = 6 \times 6$ ), to woh square side sirf ek baar count hoti hai.

6. Flashcards

n

ka factor kya hota hai?

Ek whole number

a

n

ko exactly divide kare, yaani

n = a \times k

remainder 0 ke saath.

a

ka multiple kya hota hai?

a \times k

form ka koi bhi number; agar

a

n

ka factor hai, to

n

a

ka multiple hai.

n

ka factor pair kya hota hai?

Ek pair

(a, b)

jisme

a \times b = n

n

ke saare factors dhundhne ke liye kitni value tak search karni chahiye?

⌊ n ⌋

n

search har factor kyon dhundh leta hai?

Har factor pair ka kam se kam ek member

\leq n

hota hai; dono ka

> n

hona product

> n

deta.

Perfect squares mein odd number of factors kyon hote hain?

Factor

n

khud se pair karta hai, isliye ek baar count hota hai.

Ek prime number ke kitne factors hote hain?

Exactly 2 — yaani 1 aur khud woh number.

24 ke saare factors list karo.

1, 2, 3, 4, 6, 8, 12, 24.

36 ke saare factors list karo.

1, 2, 3, 4, 6, 9, 12, 18, 36 (nau factors).

Kya 1 har number ka factor hai?

Haan, kyunki

n = 1 \times n

hamesha hota hai.

30 ke factor pairs?

(1,30), (2,15), (3,10), (5,6).

7. Connections

Prime numbers and composite numbers — factor counts inhe define karte hain.
Prime factorisation — sabse gehri factoring, blocks ke building blocks.
HCF and LCM — shared factors/multiples se directly compute hote hain.
Divisibility rules — test karne ke shortcuts ki $d$ factor hai ya nahi.
Perfect squares — odd-factor-count property.
Fractions in lowest terms — common factors cancel karo.

$d$	$24 \div d$	pair?
1	24	✅ $(1, 24)$
2	12	✅ $(2, 12)$
3	8	✅ $(3, 8)$
4	6	✅ $(4, 6)$

$d$	$36 \div d$	pair
1	36	$(1, 36)$
2	18	$(2, 18)$
3	12	$(3, 12)$
4	9	$(4, 9)$
6	6	$(6, 6)$ ← same number!