🔢(Tips, Shortcuts & Exam Tricks) – Relationships Between Numbers

 This topic mainly includes factors, multiples, LCM–HCF, divisibility, remainders, ratios of numbers, and number properties.

🔥 SSC CGL Smart Strategy

✔ Factorize quickly (2, 3, 5 first)
✔ Use logic over calculation
✔ Memorize divisibility rules
✔ Practice remainders & LCM–HCF daily
✔ Learn common factor pairs

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✅ 1. LCM–HCF Golden Rule

For two numbers:

$$\text{LCM} \times \text{HCF} = \text{Product of numbers}$$

Shortcut Use:
If one number = HCF × x
Other = HCF × y
→ LCM = HCF × x × y (when x, y are co-prime)

Example:
HCF = 12, numbers = 12 & 60
LCM = (12×60)/12 = 60

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✅ 2. Fast Method to Find Numbers When LCM & HCF Are Given

If:
HCF = h
LCM = l

Then numbers = h×x and h×y,
where x×y = l/h

Example:
HCF = 6, LCM = 180
→ l/h = 30 → x×y = 30
Possible pairs: (5,6)
Numbers = 30 & 36

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✅ 3. Divisibility Rules (SSC Favourite)

Number	Rule
3	Sum of digits divisible by 3
4	Last 2 digits divisible by 4
8	Last 3 digits divisible by 8
9	Sum of digits divisible by 9
11	(Odd sum − Even sum) multiple of 11

👉 Saves 30–40 seconds per question

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✅ 4. Number of Factors (Very Important)

If:

$$N = p^a q^b r^c$$

Then:

$$\text{Number of factors} = (a+1)(b+1)(c+1)$$

Example:
$360 = 2^3 × 3^2 × 5^1$
Factors = (3+1)(2+1)(1+1) = 24

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✅ 5. Even–Odd Logic Shortcut

• Even ± Even = Even

• Odd ± Odd = Even

• Even × Any = Even

• Odd × Odd = Odd

👉 Use logic instead of calculation.

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✅ 6. Remainder Theorem Shortcut

If number is divided by a, remainder = r, then:

• (Number + k·a) → remainder still r

• (Number × k) → remainder = (r×k) mod a

Example:
If N ÷ 7 gives remainder 3
Find remainder of 5N ÷ 7
→ (5×3) mod 7 = 1

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✅ 7. Numbers in Ratio (Most Repeated Concept)

If numbers are in ratio a : b,
Let numbers = ax and bx

Use condition to find x.

Example:
Sum = 72, ratio = 5:7
→ 12x = 72 → x = 6
Numbers = 30, 42

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✅ 8. Co-prime Numbers Trick

• Consecutive numbers → always co-prime

• HCF of co-primes = 1

• LCM = product

Example:
LCM of 14 & 15 = 14×15 = 210

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✅ 9. Perfect Square / Cube Check

• Perfect square → even number of factors

• Perfect cube → exponents multiple of 3

Example:
2⁴×3²×5¹ → not perfect square (odd power of 5)

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✅ 10. Numbers Ending with Same Digits

For numbers ending in same last digit:
Difference divisible by 10

Example:
Is 3875 and 125 divisible by 10?
→ 3875 − 125 = 3750 ✔

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