HCF LCM

महत्तम समापवर्तक (Highest Common Factor)

संख्याहरूको गुणनखण्ड (factors) सीमित हुने भएकाले, हामी संख्याहरूको ठुलो साझा गुणनखण्ड (Highest Common Factor - HCF) को बारे चर्चा गर्दछौ। HCF लाई सामान्यतया Greatest Common Divisor (GCD) वा Greatest Common Factor (GCF) पनि भनिन्छ।

1. \( \text{HCF}(1, Z) = 1 \)
2. यदि \( y \) को गुणनखण्ड \( x \) भएमा \( \text{HCF}(x, y) = x \) हुन्छ।
   उदाहरण: \( \text{HCF}(4, 8) = 4 \)
3. दुईवटा co-primes वा relatively prime संख्याहरूको HCF \( 1 \) हुन्छ।
   जस्तै \( (5,7) \), \( (14, 27) \), वा \( (11,12) \)

HCF: Let \( a \) and \( b \) are two numbers, then \( \text{HCF} \) of \( a \) and \( b \) is the highest common factor that divides the numbers \( a \) and \( b \) both without a remainder, is denoted as \( \text{HCF}(a,b) \).

Prime factor method

साझा गुणनखण्ड विधिबाट HCF पत्ता लगाउन निम्न चरणहरू अनुसरण गर्नुपर्छ।

1. प्रत्येक संख्यालाई गुणनखण्डको रूपमा लेख्नुहोस्।
2. संख्याहरूमा साझा गुणनखण्डहरु पत्ता लगाउनुहोस्।
3. साझा गुणनखण्डहरूको गुणनफल निकाल्नुहोस्।

उदाहरणका लागि, 36 र 48 को HCF पत्ता लगाउन

\( 36 = \boxed{2} \times \boxed{2} \times 3 \times \boxed{3} \)

\( 48 = \boxed{2} \times \boxed{2} \times 2 \times 2 \times \boxed{3} \)

साझा गुणनखण्डहरू \( 2, 2 \) र \( 3 \) हुन त्यसैले,

\( \text{HCF}(36, 48) = \boxed{2} \times \boxed{2} \times \boxed{3} = 12 \)

2	36,	48
2	18,	24
3	9,	12
	3,	4

Division method

भाग विधि द्वारा HCF निकाल्नको लागी ठूलो सङ्ख्यालाई सानो सङ्ख्याले भाग गर्ने र शेष (remainder) नोट गर्ने। यदि शेष शून्य छ भने, भाग गर्ने सङ्ख्या नै HCF हो। यदि बाँकी शून्य छैन भने, यो प्रक्रिया शेष शून्य नआउन्जेल दोहोर्याउने। अन्तिम शेष शुन्य आउने भाजक नै ति सङ्ख्याहरूको HCF हुन्छ।

गणितीय रूपमा: यदि दुई सङ्ख्या \( a \) र \( b \) छन् जहाँ \( a > b \), भने हामी \( a \div b \) गरेर बाँकी \( r \) निकाल्छौं। त्यसपछि \( b > r, r \ne 0 \), भएमा हामी \( b \div r \) गरेर भाग दोहोर्याउँछौं। यो प्रक्रिया शेष शून्य नआउन्जेल गरिन्छ।

जस्तै \( \text{HCF}(48, 18) \) को लागी

\( 48 \div 18 = 2 \), शेष \( = 12 \)

\( 18 \div 12 = 1 \), शेष \( = 6 \)

\( 12 \div 6 = 2 \), शेष \( = 0 \)

त्यसैले,

\( \text{HCF}(48, 18) = 6 \)

Visualization of HCF by grid

चित्र प्रयोग गरेर HCF निकाल्नको लागी सङ्ख्यालाई ग्रिडमा आयतको रुपमा देखाउनुहोस। यस भित्र, सानो सङ्ख्याको बर्ग बनाउनुहोस, फेरी बाकी सानो सङ्ख्याको बर्ग बनाउनुहोस, यसरी अन्तमा बन्ने बर्गको लम्बाई नै ति सङ्ख्याहरूको HCF हुन्छ। जस्तै

\( \text{HCF}(15,6) = 3 \)

Visualization of HCF by segment

चित्र प्रयोग गरेर HCF निकाल्नको लागी सङ्ख्यालाई ग्रिडमा आयतको रुपमा देखाउनुहोस। र बिकर्ण कोर्नुहोस। बिकर्णमा हुने भित्रि टुक्राहरुको सँख्या नै ति सङ्ख्याहरूको HCF हुन्छ। जस्तै

\( \text{HCF}(15,6) = 3 \)

Visualization of HCF by number count

चित्र प्रयोग गरेर HCF निकाल्नको लागी सङ्ख्यालाई गोटीको रुपमा देखाउनुहोस। र ठुलोबाट सानो सङ्ख्या हटाउदै जानुहोस। अन्तिममा बराबर बाकी रहेको सङ्ख्या नै ति सङ्ख्याहरूको HCF हुन्छ। जस्तै

\( \text{HCF}(15,6) = 3 \)

1. सङ्ख्यालाई गोटीको रुपमा सिधा रेखामा राख्नुहोस। 15 बाट 6 हटाउनुहोस。
   
2. बाकी सङ्ख्यालाई गोटीको रुपमा सिधा रेखामा राख्नुहोस। 9 बाट 6 हटाउनुहोस।
   
3. बाकी सङ्ख्यालाई गोटीको रुपमा सिधा रेखामा राख्नुहोस। 6 बाट 3 हटाउनुहोस।
   
4. बाकी सङ्ख्या 3 दुबैतिर बराबर भएकोले 3 नै HCF हो।
   

लघुतम समापवर्त्य (Lowest Common Multiple)

Lowest common multiples (LCM)

The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is exactly divisible by all the given numbers. To find the LCM, we use methods such as prime factorization, listing multiples, or the relation with the Highest Common Factor (HCF).

1. In the prime factorization method, we express each number as a product of its prime factors and then take the highest power of all prime factors present. The LCM is obtained by multiplying these highest power factors.
2. In the listing multiples method, we write the multiples of each number and identify the smallest common multiple.
3. The relationship between HCF and LCM is given by the formula
   \( \text{LCM}(a, b) = \dfrac{a \times b}{\text{HCF}(a, b)} \)
   which helps in quickly determining the LCM when the HCF is known.

LCM: Let \( a \) and \( b \) are two numbers, then \( \text{LCM} \) of \( a \) and \( b \) is the smallest or least positive integer that is divisible by both \( a \) and \( b \), is denoted as \( \text{LCM}(a,b) \).

For example, to find the LCM of 12 and 18, we use prime factorization

\( 12 = 2^2 \times 3^1 \)

\( 18 = 2^1 \times 3^2 \)

Taking the highest power of each prime, we get

\( 2^2 \times 3^2 = 4 \times 9 = 36 \)

so the LCM of 12 and 18 is 36.

म.स. र ल.स. (HCF and LCM)

For Q.No.8 (a) in BLE Examination

1. म.स. पत्ता लगाउनुहोस् (Find the HCF of): \( 2ab^2 \) and \( 4a^2b \) [1A]
 
   The Given Expressions are
\( 2ab^2 = \boxed{2} \cdot \boxed{a} \cdot \boxed{b} \cdot b \)
\( 4a^2b = \boxed{2} \cdot 2 \cdot \boxed{a} \cdot a \cdot \boxed{b} \)
So, HCF is
HCF = \( 2 \cdot a \cdot b \)
orHCF =\( 2ab \) 
2. म.स. निकाल्नुहोस् (Find the HCF of): \( (x + 3)(x - 5) \), \( (x - 5)(x + 6) \) [1A]
 
   The Given Expressions are
1st Expression = \( (x + 3) \cdot \boxed{(x - 5)} \)
2nd Expression = \( \boxed{(x - 5)} \cdot (x + 6) \)
So, HCF is
HCF = \( (x - 5) \) 
3. म.स. निकाल्नुहोस् (Find the HCF of): \( a^2b \), \( ab^2c \), \( a^2b^2c \) [1A]
 
   The Given Expressions are
1st Expression = \( a^2b = \boxed{a} \cdot a \cdot \boxed{b} \)
2nd Expression = \( ab^2c = \boxed{a} \cdot \boxed{b} \cdot b \cdot c \)
3rd Expression = \( a^2b^2c = \boxed{a} \cdot a \cdot \boxed{b} \cdot b \cdot c \)
So, HCF is
HCF = \( a \cdot b =ab \) 
4. म.स. पत्ता लगाउनुहोस् (Find the HCF of): \( x^2 + 7x + 10 \), \( x^2 - x - 6 \) [2A]
 
   The Given Expressions are
1st Expression = \( x^2 + 7x + 10 \)
or1st Expression = \( x^2 + 5x + 2x + 10 \)
or1st Expression = \( x(x + 5) + 2(x + 5) \)
or1st Expression = \( (x + 5) \cdot \boxed{(x + 2)} \)
Next
2nd Expression = \( x^2 - x - 6 \)
or2nd Expression =\( x^2 - 3x + 2x - 6 \)
or2nd Expression =\( x(x - 3) + 2(x - 3) \)
or2nd Expression =\( (x - 3) \cdot \boxed{(x + 2)} \)
So, HCF is
HCF = \( x + 2 \) 
5. म.स. पत्ता लगाउनुहोस् (Find the HCF of): \( 3x^3 - 15x^2 \), \( 2x^3 - 50x \) [2A]
 
   The Given Expressions are
1st Expression = \( 3x^3 - 15x^2 \)
or1st Expression = \( 3 \cdot \boxed{x} \cdot x \cdot \boxed{(x - 5)} \)
Next
2nd Expression = \( 2x^3 - 50x \)
or2nd Expression =\( 2x(x^2 - 25) \)
or2nd Expression =\( 2 \cdot \boxed{x} \cdot (x^2 - 5^2) \)
or2nd Expression =\( 2 \cdot \boxed{x} \cdot \boxed{(x - 5)} \cdot (x + 5) \)
So, HCF is
HCF = \( x(x - 5) \) 
6. म.स. पत्ता लगाउनुहोस् (Find the HCF of): \( a^2 - 4 \), \( a^2 + 5a + 6 \) [2A]
 
   The Given Expressions are
1st Expression = \( a^2 - 4 \)
or1st Expression = \( a^2 - 2^2 \)
or1st Expression = \( (a - 2) \cdot \boxed{(a + 2)} \)
Next
2nd Expression = \( a^2 + 5a + 6 \)
or2nd Expression =\( a^2 + 3a + 2a + 6 \)
or2nd Expression =\( a(a + 3) + 2(a + 3) \)
or2nd Expression =\( (a + 3) \cdot \boxed{(a + 2)} \)
So, HCF is
HCF = \( a + 2 \) 
7. म.स. निकाल्नुहोस् (Find the HCF of): \( 5x^2 - 125 \), \( x^2 - 10x + 25 \) and \( 2x^2 - 10x \) [2A]
 
   The Given Expressions are
1st Expression = \( 5x^2 - 125 \)
or1st Expression = \( 5(x^2 - 25) \)
or1st Expression = \( 5 \cdot (x + 5) \cdot \boxed{(x - 5)} \)
Next
2nd Expression = \( x^2 - 10x + 25 \)
or2nd Expression =\( x^2 - 5x - 5x + 25 \)
or2nd Expression =\( x(x - 5) - 5(x - 5) \)
or2nd Expression =\( \boxed{(x - 5)} \cdot (x - 5) \)
Next
3rd Expression = \( 2x^2 - 10x \)
or3rd Expression =\( 2x(x - 5) \)
or3rd Expression =\( 2 \cdot x \cdot \boxed{(x - 5)} \)
So, HCF is
HCF = \( x - 5 \) 
8. म.स. निकाल्नुहोस् (Find the HCF of): \( x^3 + 7x^2 + 12x \), \( x^3 + 64 \) and \( 3x^2 + 27x + 60 \) [2A]
 
   The Given Expressions are
1st Expression = \( x^3 + 7x^2 + 12x \)
or1st Expression = \( x(x^2 + 7x + 12) \)
or1st Expression = \( x(x^2 + 4x + 3x + 12) \)
or1st Expression = \( x[x(x + 4) + 3(x + 4)] \)
or1st Expression = \( x \cdot (x + 3) \cdot \boxed{(x + 4)} \)
Next
2nd Expression = \( x^3 + 64 \)
or2nd Expression =\( x^3 + 4^3 \)
or2nd Expression =\( (x + 4)(x^2 - 4x + 4^2) \)
or2nd Expression =\( \boxed{(x + 4)} \cdot (x^2 - 4x + 16) \)
Next
3rd Expression = \( 3x^2 + 27x + 60 \)
or3rd Expression =\( 3(x^2 + 9x + 20) \)
or3rd Expression =\( 3(x^2 + 5x + 4x + 20) \)
or3rd Expression =\( 3[x(x + 5) + 4(x + 5)] \)
or3rd Expression =\( 3 \cdot (x + 5) \cdot \boxed{(x + 4)} \)
So, HCF is
HCF = \( x + 4 \) 
9. म.स. पत्ता लगाउनुहोस् (Find HCF of): \( y^2 - 5y + 6 \), \( y^3 - 8 \), \( 2y^2 - 8 \) [2A]
 
   The Given Expressions are
1st Expression = \( y^2 - 5y + 6 \)
or1st Expression = \( y^2 - 3y - 2y + 6 \)
or1st Expression = \( y(y - 3) - 2(y - 3) \)
or1st Expression = \( (y - 3) \cdot \boxed{(y - 2)} \)
Next
2nd Expression = \( y^3 - 8 \)
or2nd Expression =\( y^3 - 2^3 \)
or2nd Expression =\( (y - 2)(y^2 + 2y + 4) \)
or2nd Expression =\( \boxed{(y - 2)} \cdot (y^2 + 2y + 4) \)
Next
3rd Expression = \( 2y^2 - 8 \)
or3rd Expression =\( 2(y^2 - 4) \)
or3rd Expression =\( 2(y - 2)(y + 2) \)
or3rd Expression =\( 2 \cdot \boxed{(y - 2)} \cdot (y + 2) \)
So, HCF is
HCF = \( y - 2 \) 
10. म.स. पत्ता लगाउनुहोस् (Find the HCF of): \( m^3 - 9m \), \( m^2 + m - 6 \), \( m^4 + 27m \) [2A]
 
   The Given Expressions are
1st Expression = \( m^3 - 9m \)
or1st Expression = \( m(m^2 - 9) \)
or1st Expression = \( m \cdot (m + 3) \cdot (m - 3) \)
Next
2nd Expression = \( m^2 + m - 6 \)
or2nd Expression =\( m^2 + 3m - 2m - 6 \)
or2nd Expression =\( m(m + 3) - 2(m + 3) \)
or2nd Expression =\( \boxed{(m + 3)} \cdot (m - 2) \)
Next
3rd Expression = \( m^4 + 27m \)
or3rd Expression =\( m(m^3 + 27) \)
or3rd Expression =\( m(m^3 + 3^3) \)
or3rd Expression =\( m \cdot \boxed{(m + 3)} \cdot (m^2 - 3m + 9) \)
So, HCF is
HCF = \( m + 3 \) 
11. म.स. पत्ता लगाउनुहोस् (Find the HCF of): \( 4a^3 - 9a \), \( 6a^2 + 9a \), \( 6a^3 + 5a^2 - 6a \) [2A]
 
   The Given Expressions are
1st Expression = \( 4a^3 - 9a \)
or1st Expression = \( a(4a^2 - 9) \)
or1st Expression = \( a(2a - 3) \cdot \boxed{(2a + 3)} \)
Next
2nd Expression = \( 6a^2 + 9a \)
or2nd Expression =\( 3a(2a + 3) \)
or2nd Expression =\( 3 \cdot \boxed{a} \cdot \boxed{(2a + 3)} \)
Next
3rd Expression = \( 6a^3 + 5a^2 - 6a \)
or3rd Expression =\( a(6a^2 + 5a - 6) \)
or3rd Expression =\( a(6a^2 + 9a - 4a - 6) \)
or3rd Expression =\( a[3a(2a + 3) - 2(2a + 3)] \)
or3rd Expression =\( \boxed{a} \cdot (3a - 2) \cdot \boxed{(2a + 3)} \)
So, HCF is
HCF = \( a(2a + 3) \) 
12. म.स. पत्ता लगाउनुहोस् (Find the HCF of): \( p^3 + 2p^2 - 15p \), \( p^2 - 7p + 12 \) and \( 3p^2 - 27 \) [2A]
 
   The Given Expressions are
1st Expression = \( p^3 + 2p^2 - 15p \)
or1st Expression = \( p(p^2 + 2p - 15) \)
or1st Expression = \( p(p^2 + 5p - 3p - 15) \)
or1st Expression = \( p[p(p + 5) - 3(p + 5)] \)
or1st Expression = \( p \cdot (p + 5) \cdot \boxed{(p - 3)} \)
Next
  2nd Expression = \( p^2 - 7p + 12 \)
or2nd Expression =\( p^2 - 4p - 3p + 12 \)
or2nd Expression =\( p(p - 4) - 3(p - 4) \)
or2nd Expression =\( (p - 4) \cdot \boxed{(p - 3)} \)
Next
  3rd Expression = \( 3p^2 - 27 \)
or3rd Expression =\( 3(p^2 - 9) \)
or3rd Expression =\( 3(p^2 - 3^2) \)
or3rd Expression =\( 3 \cdot (p + 3) \cdot \boxed{(p - 3)} \)
So, HCF is
HCF = \( p - 3 \) 
13. म.स. पत्ता लगाउनुहोस् (Find the HCF of): \( x^3 + 6x^2 - 4x - 24 \), \( x^2 + 5x + 6 \) and \( x^2 - 4 \) [2A]
 
   The Given Expressions are
1st Expression = \( x^3 + 6x^2 - 4x - 24 \)
or1st Expression = \( x^2(x + 6) - 4(x + 6) \)
or1st Expression = \( (x^2 - 4)(x + 6) \)
or1st Expression = \( (x - 2) \cdot \boxed{(x + 2)} \cdot (x + 6) \)
Next
  2nd Expression = \( x^2 + 5x + 6 \)
or2nd Expression =\( x^2 + 3x + 2x + 6 \)
or2nd Expression =\( x(x + 3) + 2(x + 3) \)
or2nd Expression =\( (x + 3) \cdot \boxed{(x + 2)} \)
Next
  3rd Expression = \( x^2 - 4 \)
or3rd Expression =\( x^2 - 2^2 \)
or3rd Expression =\( (x - 2) \cdot \boxed{(x + 2)} \)
So, HCF is
HCF = \( x + 2 \) 
14. म.स. पत्ता लगाउनुहोस् (Find the HCF of): \( a^2 - 25 \), \( a^2 - 6a + 5 \) and \( (a - 5)^2 \) [2A]
 
   The Given Expressions are
1st Expression = \( a^2 - 25 \)
or1st Expression = \( a^2 - 5^2 \)
or1st Expression = \( (a + 5) \cdot \boxed{(a - 5)} \)
Next
  2nd Expression = \( a^2 - 6a + 5 \)
or2nd Expression =\( a^2 - 5a - a + 5 \)
or2nd Expression =\( a(a - 5) - 1(a - 5) \)
or2nd Expression =\( (a - 1) \cdot \boxed{(a - 5)} \)
Next
  3rd Expression = \( (a - 5)^2 \)
or3rd Expression =\( \boxed{(a - 5)} \cdot (a - 5) \)
So, HCF is
HCF = \( a - 5 \) 
15. म.स. पत्ता लगाउनुहोस् (Find the HCF of): \( 2x^2 - 8 \), \( x^2 - 4x + 4 \) and \( x^2 - 3x + 2 \) [2A]
 
   The Given Expressions are
1st Expression = \( 2x^2 - 8 \)
or1st Expression = \( 2(x^2 - 4) \)
or1st Expression = \( 2(x^2 - 2^2) \)
or1st Expression = \( 2 \cdot (x + 2) \cdot \boxed{(x - 2)} \)
Next
  2nd Expression = \( x^2 - 4x + 4 \)
or2nd Expression =\( x^2 - 2x - 2x + 4 \)
or2nd Expression =\( x(x - 2) - 2(x - 2) \)
or2nd Expression =\( \boxed{(x - 2)} \cdot (x - 2) \)
Next
  3rd Expression = \( x^2 - 3x + 2 \)
or3rd Expression =\( x^2 - 2x - x + 2 \)
or3rd Expression =\( x(x - 2) - 1(x - 2) \)
or3rd Expression =\( (x - 1) \cdot \boxed{(x - 2)} \)
So, HCF is
HCF = \( x - 2 \) 
16. ल.स. पत्ता लगाउनुहोस् (Find the LCM of): \( 6ab^2 \) and \( 3ab \) [1A]
 
   The Given Expressions are
1st Expression = \( 6ab^2 = 2 \cdot 3 \cdot a \cdot b^2 \)
2nd Expression = \( 3ab = 3 \cdot a \cdot b \)
To find the LCM, we take the highest power of all factors.
So, LCM is
LCM = \( 2 \cdot 3 \cdot a \cdot b^2 = 6ab^2 \) 
17. ल.स. पत्ता लगाउनुहोस् (Find the LCM of): \( a^3b^2 \) and \( a^2b^3 \) [1A]
 
   The Given Expressions are
1st Expression = \( a^3b^2 = \boxed{a^3} \cdot b^2 \)
2nd Expression = \( a^2b^3 = a^2 \cdot \boxed{b^3} \)
To find the LCM, we take the highest power of all factors.
So, LCM is
LCM = \( \boxed{a^3} \cdot \boxed{b^3} = a^3b^3 \) 
18. ल.स. निकाल्नुहोस् (Find the LCM of): \( a + b \), \( a^2 - b^2 \) [1A]
 
   The Given Expressions are
1st Expression = \( \boxed{(a + b)} \)
2nd Expression = \( a^2 - b^2 = (a + b) \cdot \boxed{(a - b)} \)
To find the LCM, we take all the factors with their highest powers.
So, LCM is
LCM = \( (a + b)(a - b)=a^2 - b^2 \) 
19. ल.स. पत्ता लगाउनुहोस् (Find the LCM of): \( a^2 - 1 \), \( a^2 + a - 2 \) [2A]
 
   The Given Expressions are
1st Expression = \( a^2 - 1 \)
or1st Expression = \( a^2 - 1^2 \)
or1st Expression = \( \boxed{(a + 1)} \cdot \boxed{(a - 1)} \)
Next
2nd Expression = \( a^2 + a - 2 \)
or2nd Expression =\( a^2 + 2a - a - 2 \)
or2nd Expression =\( a(a + 2) - 1(a + 2) \)
or2nd Expression =\( (a - 1) \cdot \boxed{(a + 2)} \)
So, LCM is
LCM = \( (a + 1)(a - 1)(a + 2) \)
orLCM = \( (a^2 - 1)(a + 2) \) 
20. ल.स. पत्ता लगाउनुहोस् (Find the LCM of): \( x^2 + x - 20 \), \( x^2 - 25 \) [2A]
 
   The Given Expressions are
1st Expression = \( x^2 + x - 20 \)
or1st Expression = \( x^2 + 5x - 4x - 20 \)
or1st Expression = \( x(x + 5) - 4(x + 5) \)
or1st Expression = \( \boxed{(x + 5)} \cdot \boxed{(x - 4)} \)
Next
  2nd Expression = \( x^2 - 25 \)
or2nd Expression =\( x^2 - 5^2 \)
or2nd Expression =\( (x + 5) \cdot \boxed{(x - 5)} \)
So, LCM is
LCM = \( (x + 5)(x - 4)(x - 5) \)
orLCM = \( (x^2 - 25)(x - 4) \) 
21. ल.स. पत्ता लगाउनुहोस् (Find the LCM of): \( 4a - 24 \), \( a^2 - 36 \) and \( a^2 - 3a - 18 \) [3A]
 
   The Given Expressions are
1st Expression = \( 4a - 24 \)
or1st Expression = \( 4(a - 6) = \boxed{2^2} \cdot \boxed{(a - 6)} \)
Next
2nd Expression = \( a^2 - 36 \)
or2nd Expression =\( a^2 - 6^2 \)
or2nd Expression =\( (a - 6) \cdot \boxed{(a + 6)} \)
Next
3rd Expression = \( a^2 - 3a - 18 \)
or3rd Expression =\( a^2 - 6a + 3a - 18 \)
or3rd Expression =\( a(a - 6) + 3(a - 6) \)
or3rd Expression =\( (a - 6) \cdot \boxed{(a + 3)} \)
So, LCM is
LCM = \( 2^2(a - 6)(a + 6)(a + 3) \)
orLCM = \( 4(a^2 - 36)(a + 3) \) 
22. ल.स. पत्ता लगाउनुहोस् (Find the LCM of): \( (x + 2)^2 \), \( x^2 + 6x + 8 \) and \( x^2 + 7x + 10 \) [3A]
 
   The Given Expressions are
1st Expression = \( (x + 2)^2 = \boxed{(x + 2)^2} \)
Next
  2nd Expression = \( x^2 + 6x + 8 \)
or2nd Expression =\( x^2 + 4x + 2x + 8 \)
or2nd Expression =\( x(x + 4) + 2(x + 4) \)
or2nd Expression =\( \boxed{(x + 4)} \cdot (x + 2) \)
Next
  3rd Expression = \( x^2 + 7x + 10 \)
or3rd Expression =\( x^2 + 5x + 2x + 10 \)
or3rd Expression =\( x(x + 5) + 2(x + 5) \)
or3rd Expression =\( \boxed{(x + 5)} \cdot (x + 2) \)
So, LCM is
LCM = \( (x + 2)^2(x + 4)(x + 5) \) 
23. ल.स. पत्ता लगाउनुहोस् (Find the LCM of): \( x^2 + 3x - 10 \), \( x^2 - 6x + 8 \) and \( x^2 + x - 20 \) [3A]
 
   The Given Expressions are
1st Expression = \( x^2 + 3x - 10 \)
or1st Expression = \( x^2 + 5x - 2x - 10 \)
or1st Expression = \( x(x + 5) - 2(x + 5) \)
or1st Expression = \( \boxed{(x - 2)} \cdot \boxed{(x + 5)} \)
Next
2nd Expression = \( x^2 - 6x + 8 \)
or2nd Expression =\( x^2 - 4x - 2x + 8 \)
or2nd Expression =\( x(x - 4) - 2(x - 4) \)
or2nd Expression =\((x - 2) \cdot \boxed{(x - 4)} \)
Next
3rd Expression = \( x^2 + x - 20 \)
or3rd Expression =\( x^2 + 5x - 4x - 20 \)
or3rd Expression =\( x(x + 5) - 4(x + 5) \)
or3rd Expression =\( (x + 5) \cdot (x - 4) \)
So, LCM is
LCM = \( (x - 2)(x + 5)(x - 4) \) 
24. ल.स. पत्ता लगाउनुहोस् (Find the LCM of): \( a^2 + 6a + 8 \), \( a^2 + 9a + 20 \) and \( a^2 + 7a + 10 \) [3A]
 
   The Given Expressions are
1st Expression = \( a^2 + 6a + 8 \)
or1st Expression = \( a^2 + 4a + 2a + 8 \)
or1st Expression = \( a(a + 4) + 2(a + 4) \)
or1st Expression = \( \boxed{(a + 4)} \cdot \boxed{(a + 2)} \)
Next
2nd Expression = \( a^2 + 9a + 20 \)
or2nd Expression =\( a^2 + 5a + 4a + 20 \)
or2nd Expression =\( a(a + 5) + 4(a + 5) \)
or2nd Expression =\( \boxed{(a + 5)} \cdot (a + 4) \)
Next
3rd Expression = \( a^2 + 7a + 10 \)
or3rd Expression =\( a^2 + 5a + 2a + 10 \)
or3rd Expression =\( a(a + 5) + 2(a + 5) \)
or3rd Expression =\( (a + 5) \cdot(a + 2) \)
So, LCM is
LCM = \( (a + 4)(a + 2)(a + 5) \) 
25. म.स. र ल.स. पत्ता लगाउनुहोस् (Find the HCF and LCM of): \( x^2 - 5x + 6 \) and \( x^3 - 4x \) [3A]
 
   The Given Expressions are
1st Expression = \( x^2 - 5x + 6 \)
or1st Expression = \( x^2 - 3x - 2x + 6 \)
or1st Expression = \( x(x - 3) - 2(x - 3) \)
or1st Expression = \( \boxed{(x - 2)} \cdot (x - 3) \)
Next
2nd Expression = \( x^3 - 4x \)
or2nd Expression = \( x(x^2 - 4) \)
or2nd Expression = \( x \cdot (x^2 - 2^2) \)
or2nd Expression = \( x \cdot \boxed{(x - 2)} \cdot (x + 2) \)
So, HCF is
HCF = \( x - 2 \)
Next, LCM is
LCM = \( (x - 2)(x - 3) x(x + 2) \)
orLCM = \( x(x^2 - 4)(x - 3) \) 
26. म.स. र ल.स. पत्ता लगाउनुहोस् (Find the HCF and LCM of): \( x^2 - 25 \) and \( x^2 - 9x + 20 \) [3A]
 
   The Given Expressions are
1st Expression = \( x^2 - 25 \)
or1st Expression = \( x^2 - 5^2 \)
or1st Expression = \( (x + 5) \cdot \boxed{(x - 5)} \)
Next
2nd Expression = \( x^2 - 9x + 20 \)
or2nd Expression = \( x^2 - 5x - 4x + 20 \)
or2nd Expression = \( x(x - 5) - 4(x - 5) \)
or2nd Expression = \( (x - 4) \cdot \boxed{(x - 5)} \)
  So, HCF is
HCF = \( x - 5 \)
Next, LCM is
LCM = \( (x + 5) \cdot (x - 5) \cdot (x - 4) \)
orLCM = \( (x^2 - 25)(x - 4) \) 
27. म.स. र ल.स. पत्ता लगाउनुहोस् (Find the HCF and LCM of): \( x^2 + x - 6 \) and \( x^2 - 9 \) [3A]
 
   The Given Expressions are
1st Expression = \( x^2 + x - 6 \)
or1st Expression = \( x^2 + 3x - 2x - 6 \)
or1st Expression = \( x(x + 3) - 2(x + 3) \)
or1st Expression = \( \boxed{(x + 3)} \cdot (x - 2) \)
Next
2nd Expression = \( x^2 - 9 \)
or2nd Expression = \( x^2 - 3^2 \)
or2nd Expression = \( \boxed{(x + 3)} \cdot (x - 3) \)
  So, HCF is
HCF = \( x + 3 \)
Next, LCM is
LCM = \( (x + 3)(x - 2)(x - 3) \)
orLCM = \( (x - 2)(x^2 - 9) \) 
28. म.स. र ल.स. पत्ता लगाउनुहोस् (Find the HCF and LCM of): \( x^2 - z^2 + y^2 + 2xy \) and \( x^2 - y^2 + z^2 - 2xz \) [3A]
 
   The Given Expressions are
1st Expression = \( x^2 - z^2 + y^2 + 2xy \)
or1st Expression = \( (x^2 + 2xy + y^2) - z^2 \)
or1st Expression = \( (x + y)^2 - z^2 \)
or1st Expression = \( \boxed{(x + y + z)} \cdot (x + y - z) \)
Next
2nd Expression = \( x^2 - y^2 + z^2 - 2xz \)
or2nd Expression = \( (x^2 - 2xz + z^2) - y^2 \)
or2nd Expression = \( (x - z)^2 - y^2 \)
or2nd Expression = \( [(x - z) + y] \cdot [(x - z) - y] \)
or2nd Expression = \( \boxed{(x + y - z)} \cdot (x - y - z) \)
  So, HCF is
HCF = \( x + y - z \)
Next, LCM is
LCM = \( (x + y - z) \cdot (x + y + z) \cdot (x - y - z) \)