Q1 (Asked in 2024):

Use Euclid’s Division Algorithm to find the HCF of 135 and 225.

Solution: Step 1: 225 = 135 × 1 + 90. Step 2: 135 = 90 × 1 + 45. Step 3: 90 = 45 × 2 + 0. Since remainder is 0, HCF = 45.

Formula Used: a = bq + r, where 0 ≤ r < b

Q2 (Asked in 2023):

Prove that √2 is irrational.

Solution: Assume √2 = p/q, where p, q are coprime integers, q ≠ 0. Square both sides: 2 = p²/q², so p² = 2q². Thus, p² is even, so p is even (p = 2m). Then, 4m² = 2q², so q² = 2m², implying q is even. Since p and q are both even, they share a common factor, contradicting coprimality. Hence, √2 is irrational.

Formula Used: Irrational numbers cannot be expressed as p/q

Q3 (Asked in 2022):

Find the LCM of 12 and 18 using prime factorization.

Solution: Prime factorization: 12 = 2² × 3, 18 = 2 × 3². LCM = 2² × 3² = 4 × 9 = 36.

Formula Used: LCM = highest powers of all prime factors

Q4 (Asked in 2021):

Express 156 as a product of its prime factors.

Solution: Divide 156: 156 ÷ 2 = 78, 78 ÷ 2 = 39, 39 ÷ 3 = 13, 13 ÷ 13 = 1. Thus, 156 = 2² × 3 × 13.

Formula Used: Fundamental Theorem of Arithmetic

Q5 (Asked in 2020):

Find the HCF of 96 and 404 using Euclid’s algorithm.

Solution: Step 1: 404 = 96 × 4 + 20. Step 2: 96 = 20 × 4 + 16. Step 3: 20 = 16 × 1 + 4. Step 4: 16 = 4 × 4 + 0. HCF = 4.

Formula Used: a = bq + r

Q6 (Asked in 2019):

Show that 3 + 2√5 is irrational.

Solution: Assume 3 + 2√5 = p/q, where p, q are coprime, q ≠ 0. Then, 2√5 = p/q – 3, so √5 = (p – 3q)/(2q). Since p, q are integers, (p – 3q)/(2q) is rational. But √5 is irrational, a contradiction. Hence, 3 + 2√5 is irrational.

Formula Used: Sum of rational and irrational is irrational

Q7 (Asked in 2018):

Find the LCM and HCF of 15 and 25.

Solution: Prime factorization: 15 = 3 × 5, 25 = 5². LCM = 3 × 5² = 75. HCF = 5. Verify: LCM × HCF = 75 × 5 = 375 = 15 × 25.

Formula Used: HCF(a, b) × LCM(a, b) = a × b

Q8 (Asked in 2017):

Express 3825 as a product of prime factors.

Solution: Divide 3825: 3825 ÷ 5 = 765, 765 ÷ 5 = 153, 153 ÷ 3 = 51, 51 ÷ 3 = 17, 17 ÷ 17 = 1. Thus, 3825 = 3² × 5² × 17.

Formula Used: Fundamental Theorem of Arithmetic

Q9 (Asked in 2016):

Find the HCF of 65 and 117 using Euclid’s algorithm.

Solution: Step 1: 117 = 65 × 1 + 52. Step 2: 65 = 52 × 1 + 13. Step 3: 52 = 13 × 4 + 0. HCF = 13.

Formula Used: a = bq + r

Q10 (Asked in 2015):

Prove that √3 is irrational.

Solution: Assume √3 = p/q, where p, q are coprime, q ≠ 0. Then, 3 = p²/q², so p² = 3q². Thus, p² is divisible by 3, so p is divisible by 3 (p = 3m). Then, 9m² = 3q², so q² = 3m². Thus, q is divisible by 3, contradicting coprimality. Hence, √3 is irrational.

Formula Used: Irrational numbers cannot be expressed as p/q

Q11 (Asked in 2014):

Find the LCM of 8 and 12 using prime factorization.

Solution: Prime factorization: 8 = 2³, 12 = 2² × 3. LCM = 2³ × 3 = 24.

Formula Used: LCM = highest powers of all prime factors

Q12 (Asked in 2013):

Show that 5 – √2 is irrational.

Solution: Assume 5 – √2 = p/q, where p, q are coprime, q ≠ 0. Then, √2 = 5 – p/q = (5q – p)/q. Since (5q – p)/q is rational and √2 is irrational, a contradiction arises. Hence, 5 – √2 is irrational.

Formula Used: Difference of rational and irrational is irrational

Q13 (Asked in 2012):

Find the HCF of 81 and 237 using Euclid’s algorithm.

Solution: Step 1: 237 = 81 × 2 + 75. Step 2: 81 = 75 × 1 + 6. Step 3: 75 = 6 × 12 + 3. Step 4: 6 = 3 × 2 + 0. HCF = 3.

Formula Used: a = bq + r

Q14 (Asked in 2011):

Express 7429 as a product of its prime factors.

Solution: Divide 7429: 7429 ÷ 17 = 437, 437 ÷ 19 = 23, 23 ÷ 23 = 1. Thus, 7429 = 17 × 19 × 23.

Formula Used: Fundamental Theorem of Arithmetic

Q15 (Asked in 2010):

Find the LCM of 20 and 30.

Solution: Prime factorization: 20 = 2² × 5, 30 = 2 × 3 × 5. LCM = 2² × 3 × 5 = 60.

Formula Used: LCM = highest powers of all prime factors

Q16 (Asked in 2024):

Prove that √5 is irrational.

Solution: Assume √5 = p/q, where p, q are coprime, q ≠ 0. Then, 5 = p²/q², so p² = 5q². Thus, p² is divisible by 5, so p is divisible by 5 (p = 5m). Then, 25m² = 5q², so q² = 5m². Thus, q is divisible by 5, contradicting coprimality. Hence, √5 is irrational.

Formula Used: Irrational numbers cannot be expressed as p/q

Q17 (Asked in 2023):

Find the HCF of 56 and 98 using Euclid’s algorithm.

Solution: Step 1: 98 = 56 × 1 + 42. Step 2: 56 = 42 × 1 + 14. Step 3: 42 = 14 × 3 + 0. HCF = 14.

Formula Used: a = bq + r

Q18 (Asked in 2022):

Express 2028 as a product of its prime factors.

Solution: Divide 2028: 2028 ÷ 2 = 1014, 1014 ÷ 2 = 507, 507 ÷ 3 = 169, 169 ÷ 13 = 13, 13 ÷ 13 = 1. Thus, 2028 = 2² × 3 × 13².

Formula Used: Fundamental Theorem of Arithmetic

Q19 (Asked in 2021):

Find the LCM of 36 and 48.

Solution: Prime factorization: 36 = 2² × 3², 48 = 2⁴ × 3. LCM = 2⁴ × 3² = 16 × 9 = 144.

Formula Used: LCM = highest powers of all prime factors

Q20 (Asked in 2020):

Show that 7 – √3 is irrational.

Solution: Assume 7 – √3 = p/q, where p, q are coprime, q ≠ 0. Then, √3 = 7 – p/q = (7q – p)/q. Since (7q – p)/q is rational and √3 is irrational, a contradiction arises. Hence, 7 – √3 is irrational.

Formula Used: Difference of rational and irrational is irrational

Q21 (Asked in 2019):

Find the HCF of 72 and 126 using Euclid’s algorithm.

Solution: Step 1: 126 = 72 × 1 + 54. Step 2: 72 = 54 × 1 + 18. Step 3: 54 = 18 × 3 + 0. HCF = 18.

Formula Used: a = bq + r

Q22 (Asked in 2018):

Express 2310 as a product of its prime factors.

Solution: Divide 2310: 2310 ÷ 2 = 1155, 1155 ÷ 3 = 385, 385 ÷ 5 = 77, 77 ÷ 7 = 11, 11 ÷ 11 = 1. Thus, 2310 = 2 × 3 × 5 × 7 × 11.

Formula Used: Fundamental Theorem of Arithmetic

Q23 (Asked in 2017):

Find the LCM of 16 and 24.

Solution: Prime factorization: 16 = 2⁴, 24 = 2³ × 3. LCM = 2⁴ × 3 = 48.

Formula Used: LCM = highest powers of all prime factors

Q24 (Asked in 2016):

Prove that √7 is irrational.

Solution: Assume √7 = p/q, where p, q are coprime, q ≠ 0. Then, 7 = p²/q², so p² = 7q². Thus, p² is divisible by 7, so p is divisible by 7 (p = 7m). Then, 49m² = 7q², so q² = 7m². Thus, q is divisible by 7, contradicting coprimality. Hence, √7 is irrational.

Formula Used: Irrational numbers cannot be expressed as p/q

Q25 (Asked in 2015):

Find the HCF of 144 and 198 using Euclid’s algorithm.

Solution: Step 1: 198 = 144 × 1 + 54. Step 2: 144 = 54 × 2 + 36. Step 3: 54 = 36 × 1 + 18. Step 4: 36 = 18 × 2 + 0. HCF = 18.

Formula Used: a = bq + r

Q26 (Asked in 2014):

Express 4096 as a product of its prime factors.

Solution: Divide 4096: 4096 ÷ 2 = 2048, 2048 ÷ 2 = 1024, 1024 ÷ 2 = 512, 512 ÷ 2 = 256, 256 ÷ 2 = 128, 128 ÷ 2 = 64, 64 ÷ 2 = 32, 32 ÷ 2 = 16, 16 ÷ 2 = 8, 8 ÷ 2 = 4, 4 ÷ 2 = 2, 2 ÷ 2 = 1. Thus, 4096 = 2¹².

Formula Used: Fundamental Theorem of Arithmetic

Q27 (Asked in 2013):

Find the LCM of 28 and 42.

Solution: Prime factorization: 28 = 2² × 7, 42 = 2 × 3 × 7. LCM = 2² × 3 × 7 = 84.

Formula Used: LCM = highest powers of all prime factors

Q28 (Asked in 2012):

Show that 2 + √3 is irrational.

Solution: Assume 2 + √3 = p/q, where p, q are coprime, q ≠ 0. Then, √3 = p/q – 2 = (p – 2q)/q. Since (p – 2q)/q is rational and √3 is irrational, a contradiction arises. Hence, 2 + √3 is irrational.

Formula Used: Sum of rational and irrational is irrational

Q29 (Asked in 2011):

Find the HCF of 255 and 867 using Euclid’s algorithm.

Solution: Step 1: 867 = 255 × 3 + 102. Step 2: 255 = 102 × 2 + 51. Step 3: 102 = 51 × 2 + 0. HCF = 51.

Formula Used: a = bq + r

Q30 (Asked in 2010):

Express 980 as a product of its prime factors.

Solution: Divide 980: 980 ÷ 2 = 490, 490 ÷ 2 = 245, 245 ÷ 5 = 49, 49 ÷ 7 = 7, 7 ÷ 7 = 1. Thus, 980 = 2² × 5 × 7².

Formula Used: Fundamental Theorem of Arithmetic

Q31 (Asked in 2024):

Find the LCM of 45 and 75.

Solution: Prime factorization: 45 = 3² × 5, 75 = 3 × 5². LCM = 3² × 5² = 9 × 25 = 225.

Formula Used: LCM = highest powers of all prime factors

Q32 (Asked in 2023):

Prove that √11 is irrational.

Solution: Assume √11 = p/q, where p, q are coprime, q ≠ 0. Then, 11 = p²/q², so p² = 11q². Thus, p² is divisible by 11, so p is divisible by 11 (p = 11m). Then, 121m² = 11q², so q² = 11m². Thus, q is divisible by 11, contradicting coprimality. Hence, √11 is irrational.

Formula Used: Irrational numbers cannot be expressed as p/q

Q33 (Asked in 2022):

Find the HCF of 91 and 112 using Euclid’s algorithm.

Solution: Step 1: 112 = 91 × 1 + 21. Step 2: 91 = 21 × 4 + 7. Step 3: 21 = 7 × 3 + 0. HCF = 7.

Formula Used: a = bq + r

Q34 (Asked in 2021):

Express 720 as a product of its prime factors.

Solution: Divide 720: 720 ÷ 2 = 360, 360 ÷ 2 = 180, 180 ÷ 2 = 90, 90 ÷ 2 = 45, 45 ÷ 3 = 15, 15 ÷ 3 = 5, 5 ÷ 5 = 1. Thus, 720 = 2⁴ × 3² × 5.

Formula Used: Fundamental Theorem of Arithmetic

Q35 (Asked in 2020):

Show that 4 + √5 is irrational.

Solution: Assume 4 + √5 = p/q, where p, q are coprime, q ≠ 0. Then, √5 = p/q – 4 = (p – 4q)/q. Since (p – 4q)/q is rational and √5 is irrational, a contradiction arises. Hence, 4 + √5 is irrational.

Formula Used: Sum of rational and irrational is irrational

Q36 (Asked in 2019):

Find the LCM of 50 and 75.

Solution: Prime factorization: 50 = 2 × 5², 75 = 3 × 5². LCM = 2 × 3 × 5² = 150.

Formula Used: LCM = highest powers of all prime factors

Q37 (Asked in 2018):

Find the HCF of 180 and 252 using Euclid’s algorithm.

Solution: Step 1: 252 = 180 × 1 + 72. Step 2: 180 = 72 × 2 + 36. Step 3: 72 = 36 × 2 + 0. HCF = 36.

Formula Used: a = bq + r

Q38 (Asked in 2017):

Express 1365 as a product of its prime factors.

Solution: Divide 1365: 1365 ÷ 5 = 273, 273 ÷ 3 = 91, 91 ÷ 7 = 13, 13 ÷ 13 = 1. Thus, 1365 = 3 × 5 × 7 × 13.

Formula Used: Fundamental Theorem of Arithmetic

Q39 (Asked in 2016):

Prove that √13 is irrational.

Solution: Assume √13 = p/q, where p, q are coprime, q ≠ 0. Then, 13 = p²/q², so p² = 13q². Thus, p² is divisible by 13, so p is divisible by 13 (p = 13m). Then, 169m² = 13q², so q² = 13m². Thus, q is divisible by 13, contradicting coprimality. Hence, √13 is irrational.

Formula Used: Irrational numbers cannot be expressed as p/q

Q40 (Asked in 2015):

Find the LCM of 60 and 84.

Solution: Prime factorization: 60 = 2² × 3 × 5, 84 = 2² × 3 × 7. LCM = 2² × 3 × 5 × 7 = 420.

Formula Used: LCM = highest powers of all prime factors

Q41 (Asked in 2014):

Find the HCF of 210 and 308 using Euclid’s algorithm.

Solution: Step 1: 308 = 210 × 1 + 98. Step 2: 210 = 98 × 2 + 14. Step 3: 98 = 14 × 7 + 0. HCF = 14.

Formula Used: a = bq + r

Q42 (Asked in 2013):

Express 504 as a product of its prime factors.

Solution: Divide 504: 504 ÷ 2 = 252, 252 ÷ 2 = 126, 126 ÷ 2 = 63, 63 ÷ 3 = 21, 21 ÷ 3 = 7, 7 ÷ 7 = 1. Thus, 504 = 2³ × 3² × 7.

Formula Used: Fundamental Theorem of Arithmetic

Q43 (Asked in 2012):

Show that 6 – √2 is irrational.

Solution: Assume 6 – √2 = p/q, where p, q are coprime, q ≠ 0. Then, √2 = 6 – p/q = (6q – p)/q. Since (6q – p)/q is rational and √2 is irrational, a contradiction arises. Hence, 6 – √2 is irrational.

Formula Used: Difference of rational and irrational is irrational

Q44 (Asked in 2011):

Find the LCM of 35 and 49.

Solution: Prime factorization: 35 = 5 × 7, 49 = 7². LCM = 5 × 7² = 245.

Formula Used: LCM = highest powers of all prime factors

Q45 (Asked in 2010):

Find the HCF of 135 and 315 using Euclid’s algorithm.

Solution: Step 1: 315 = 135 × 2 + 45. Step 2: 135 = 45 × 3 + 0. HCF = 45.

Formula Used: a = bq + r

Q46 (Asked in 2024):

Express 1296 as a product of its prime factors.

Solution: Divide 1296: 1296 ÷ 2 = 648, 648 ÷ 2 = 324, 324 ÷ 2 = 162, 162 ÷ 2 = 81, 81 ÷ 3 = 27, 27 ÷ 3 = 9, 9 ÷ 3 = 3, 3 ÷ 3 = 1. Thus, 1296 = 2⁴ × 3⁴.

Formula Used: Fundamental Theorem of Arithmetic

Q47 (Asked in 2023):

Prove that √17 is irrational.

Solution: Assume √17 = p/q, where p, q are coprime, q ≠ 0. Then, 17 = p²/q², so p² = 17q². Thus, p² is divisible by 17, so p is divisible by 17 (p = 17m). Then, 289m² = 17q², so q² = 17m². Thus, q is divisible by 17, contradicting coprimality. Hence, √17 is irrational.

Formula Used: Irrational numbers cannot be expressed as p/q

Q48 (Asked in 2022):

Find the LCM of 27 and 81.

Solution: Prime factorization: 27 = 3³, 81 = 3⁴. LCM = 3⁴ = 81.

Formula Used: LCM = highest powers of all prime factors

Q49 (Asked in 2021):

Find the HCF of 204 and 340 using Euclid’s algorithm.

Solution: Step 1: 340 = 204 × 1 + 136. Step 2: 204 = 136 × 1 + 68. Step 3: 136 = 68 × 2 + 0. HCF = 68.

Formula Used: a = bq + r

Q50 (Asked in 2020):

Express 1560 as a product of its prime factors.

Solution: Divide 1560: 1560 ÷ 2 = 780, 780 ÷ 2 = 390, 390 ÷ 2 = 195, 195 ÷ 3 = 65, 65 ÷ 5 = 13, 13 ÷ 13 = 1. Thus, 1560 = 2³ × 3 × 5 × 13.

Formula Used: Fundamental Theorem of Arithmetic

Q51 (Asked in 2019):

Show that 3 + √7 is irrational.

Solution: Assume 3 + √7 = p/q, where p, q are coprime, q ≠ 0. Then, √7 = p/q – 3 = (p – 3q)/q. Since (p – 3q)/q is rational and √7 is irrational, a contradiction arises. Hence, 3 + √7 is irrational.

Formula Used: Sum of rational and irrational is irrational

Q52 (Asked in 2018):

Find the LCM of 40 and 56.

Solution: Prime factorization: 40 = 2³ × 5, 56 = 2³ × 7. LCM = 2³ × 5 × 7 = 280.

Formula Used: LCM = highest powers of all prime factors

Q53 (Asked in 2017):

Find the HCF of 165 and 385 using Euclid’s algorithm.

Solution: Step 1: 385 = 165 × 2 + 55. Step 2: 165 = 55 × 3 + 0. HCF = 55.

Formula Used: a = bq + r

Q54 (Asked in 2016):

Express 2520 as a product of its prime factors.

Solution: Divide 2520: 2520 ÷ 2 = 1260, 1260 ÷ 2 = 630, 630 ÷ 2 = 315, 315 ÷ 3 = 105, 105 ÷ 3 = 35, 35 ÷ 5 = 7, 7 ÷ 7 = 1. Thus, 2520 = 2³ × 3² × 5 × 7.

Formula Used: Fundamental Theorem of Arithmetic

Q55 (Asked in 2015):

Prove that √19 is irrational.

Solution: Assume √19 = p/q, where p, q are coprime, q ≠ 0. Then, 19 = p²/q², so p² = 19q². Thus, p² is divisible by 19, so p is divisible by 19 (p = 19m). Then, 361m² = 19q², so q² = 19m². Thus, q is divisible by 19, contradicting coprimality. Hence, √19 is irrational.

Formula Used: Irrational numbers cannot be expressed as p/q

Q56 (Asked in 2014):

Find the LCM of 32 and 48.

Solution: Prime factorization: 32 = 2⁵, 48 = 2⁴ × 3. LCM = 2⁵ × 3 = 96.

Formula Used: LCM = highest powers of all prime factors

Q57 (Asked in 2013):

Find the HCF of 288 and 360 using Euclid’s algorithm.

Solution: Step 1: 360 = 288 × 1 + 72. Step 2: 288 = 72 × 4 + 0. HCF = 72.

Formula Used: a = bq + r

Q58 (Asked in 2012):

Express 1764 as a product of its prime factors.

Solution: Divide 1764: 1764 ÷ 2 = 882, 882 ÷ 2 = 441, 441 ÷ 3 = 147, 147 ÷ 3 = 49, 49 ÷ 7 = 7, 7 ÷ 7 = 1. Thus, 1764 = 2² × 3² × 7².

Formula Used: Fundamental Theorem of Arithmetic

Q59 (Asked in 2011):

Show that 5 + √3 is irrational.

Solution: Assume 5 + √3 = p/q, where p, q are coprime, q ≠ 0. Then, √3 = p/q – 5 = (p – 5q)/q. Since (p – 5q)/q is rational and √3 is irrational, a contradiction arises. Hence, 5 + √3 is irrational.

Formula Used: Sum of rational and irrational is irrational

Q60 (Asked in 2010):

Find the LCM of 18 and 27.

Solution: Prime factorization: 18 = 2 × 3², 27 = 3³. LCM = 2 × 3³ = 54.

Formula Used: LCM = highest powers of all prime factors

Q61 (Asked in 2024):

Find the HCF of 196 and 420 using Euclid’s algorithm.

Solution: Step 1: 420 = 196 × 2 + 28. Step 2: 196 = 28 × 7 + 0. HCF = 28.

Formula Used: a = bq + r

Q62 (Asked in 2023):

Express 3375 as a product of its prime factors.

Solution: Divide 3375: 3375 ÷ 5 = 675, 675 ÷ 5 = 135, 135 ÷ 5 = 27, 27 ÷ 3 = 9, 9 ÷ 3 = 3, 3 ÷ 3 = 1. Thus, 3375 = 3³ × 5³.

Formula Used: Fundamental Theorem of Arithmetic

Q63 (Asked in 2022):

Prove that √23 is irrational.

Solution: Assume √23 = p/q, where p, q are coprime, q ≠ 0. Then, 23 = p²/q², so p² = 23q². Thus, p² is divisible by 23, so p is divisible by 23 (p = 23m). Then, 529m² = 23q², so q² = 23m². Thus, q is divisible by 23, contradicting coprimality. Hence, √23 is irrational.

Formula Used: Irrational numbers cannot be expressed as p/q

Q64 (Asked in 2021):

Find the LCM of 64 and 80.

Solution: Prime factorization: 64 = 2⁶, 80 = 2⁴ × 5. LCM = 2⁶ × 5 = 320.

Formula Used: LCM = highest powers of all prime factors

Q65 (Asked in 2020):

Find the HCF of 150 and 225 using Euclid’s algorithm.

Solution: Step 1: 225 = 150 × 1 + 75. Step 2: 150 = 75 × 2 + 0. HCF = 75.

Formula Used: a = bq + r

Q66 (Asked in 2019):

Express 1080 as a product of its prime factors.

Solution: Divide 1080: 1080 ÷ 2 = 540, 540 ÷ 2 = 270, 270 ÷ 2 = 135, 135 ÷ 3 = 45, 45 ÷ 3 = 15, 15 ÷ 3 = 5, 5 ÷ 5 = 1. Thus, 1080 = 2³ × 3³ × 5.

Formula Used: Fundamental Theorem of Arithmetic

Q67 (Asked in 2018):

Show that 8 – √5 is irrational.

Solution: Assume 8 – √5 = p/q, where p, q are coprime, q ≠ 0. Then, √5 = 8 – p/q = (8q – p)/q. Since (8q – p)/q is rational and √5 is irrational, a contradiction arises. Hence, 8 – √5 is irrational.

Formula Used: Difference of rational and irrational is irrational

Q68 (Asked in 2017):

Find the LCM of 21 and 63.

Solution: Prime factorization: 21 = 3 × 7, 63 = 3² × 7. LCM = 3² × 7 = 63.

Formula Used: LCM = highest powers of all prime factors

Q69 (Asked in 2016):

Find the HCF of 132 and 220 using Euclid’s algorithm.

Solution: Step 1: 220 = 132 × 1 + 88. Step 2: 132 = 88 × 1 + 44. Step 3: 88 = 44 × 2 + 0. HCF = 44.

Formula Used: a = bq + r

Q70 (Asked in 2015):

Express 2160 as a product of its prime factors.

Solution: Divide 2160: 2160 ÷ 2 = 1080, 1080 ÷ 2 = 540, 540 ÷ 2 = 270, 270 ÷ 2 = 135, 135 ÷ 3 = 45, 45 ÷ 3 = 15, 15 ÷ 3 = 5, 5 ÷ 5 = 1. Thus, 2160 = 2⁴ × 3³ × 5.

Formula Used: Fundamental Theorem of Arithmetic

Q71 (Asked in 2014):

Prove that √29 is irrational.

Solution: Assume √29 = p/q, where p, q are coprime, q ≠ 0. Then, 29 = p²/q², so p² = 29q². Thus, p² is divisible by 29, so p is divisible by 29 (p = 29m). Then, 841m² = 29q², so q² = 29m². Thus, q is divisible by 29, contradicting coprimality. Hence, √29 is irrational.

Formula Used: Irrational numbers cannot be expressed as p/q

Q72 (Asked in 2013):

Find the LCM of 24 and 36.

Solution: Prime factorization: 24 = 2³ × 3, 36 = 2² × 3². LCM = 2³ × 3² = 72.

Formula Used: LCM = highest powers of all prime factors

Q73 (Asked in 2012):

Find the HCF of 168 and 280 using Euclid’s algorithm.

Solution: Step 1: 280 = 168 × 1 + 112. Step 2: 168 = 112 × 1 + 56. Step 3: 112 = 56 × 2 + 0. HCF = 56.

Formula Used: a = bq + r

Q74 (Asked in 2011):

Express 3600 as a product of its prime factors.

Solution: Divide 3600: 3600 ÷ 2 = 1800, 1800 ÷ 2 = 900, 900 ÷ 2 = 450, 450 ÷ 2 = 225, 225 ÷ 3 = 75, 75 ÷ 3 = 25, 25 ÷ 5 = 5, 5 ÷ 5 = 1. Thus, 3600 = 2⁴ × 3² × 5².

Formula Used: Fundamental Theorem of Arithmetic

Q75 (Asked in 2010):

Show that 9 – √7 is irrational.

Solution: Assume 9 – √7 = p/q, where p, q are coprime, q ≠ 0. Then, √7 = 9 – p/q = (9q – p)/q. Since (9q – p)/q is rational and √7 is irrational, a contradiction arises. Hence, 9 – √7 is irrational.

Formula Used: Difference of rational and irrational is irrational

Q76 (Asked in 2024):

Find the LCM of 54 and 72.

Solution: Prime factorization: 54 = 2 × 3³, 72 = 2³ × 3². LCM = 2³ × 3³ = 216.

Formula Used: LCM = highest powers of all prime factors

Q77 (Asked in 2023):

Find the HCF of 117 and 221 using Euclid’s algorithm.

Solution: Step 1: 221 = 117 × 1 + 104. Step 2: 117 = 104 × 1 + 13. Step 3: 104 = 13 × 8 + 0. HCF = 13.

Formula Used: a = bq + r

Q78 (Asked in 2022):

Express 405 as a product of its prime factors.

Solution: Divide 405: 405 ÷ 5 = 81, 81 ÷ 3 = 27, 27 ÷ 3 = 9, 9 ÷ 3 = 3, 3 ÷ 3 = 1. Thus, 405 = 3⁴ × 5.

Formula Used: Fundamental Theorem of Arithmetic

Q79 (Asked in 2021):

Prove that √31 is irrational.

Solution: Assume √31 = p/q, where p, q are coprime, q ≠ 0. Then, 31 = p²/q², so p² = 31q². Thus, p² is divisible by 31, so p is divisible by 31 (p = 31m). Then, 961m² = 31q², so q² = 31m². Thus, q is divisible by 31, contradicting coprimality. Hence, √31 is irrational.

Formula Used: Irrational numbers cannot be expressed as p/q

Q80 (Asked in 2020):

Find the LCM of 30 and 45.

Solution: Prime factorization: 30 = 2 × 3 × 5, 45 = 3² × 5. LCM = 2 × 3² × 5 = 90.

Formula Used: LCM = highest powers of all prime factors

Q81 (Asked in 2019):

Find the HCF of 175 and 245 using Euclid’s algorithm.

Solution: Step 1: 245 = 175 × 1 + 70. Step 2: 175 = 70 × 2 + 35. Step 3: 70 = 35 × 2 + 0. HCF = 35.

Formula Used: a = bq + r

Q82 (Asked in 2018):

Express 2880 as a product of its prime factors.

Solution: Divide 2880: 2880 ÷ 2 = 1440, 1440 ÷ 2 = 720, 720 ÷ 2 = 360, 360 ÷ 2 = 180, 180 ÷ 2 = 90, 90 ÷ 2 = 45, 45 ÷ 3 = 15, 15 ÷ 3 = 5, 5 ÷ 5 = 1. Thus, 2880 = 2⁷ × 3² × 5.

Formula Used: Fundamental Theorem of Arithmetic

Q83 (Asked in 2017):

Show that 7 + √2 is irrational.

Solution: Assume 7 + √2 = p/q, where p, q are coprime, q ≠ 0. Then, √2 = p/q – 7 = (p – 7q)/q. Since (p – 7q)/q is rational and √2 is irrational, a contradiction arises. Hence, 7 + √2 is irrational.

Formula Used: Sum of rational and irrational is irrational

Q84 (Asked in 2016):

Find the LCM of 15 and 35.

Solution: Prime factorization: 15 = 3 × 5, 35 = 5 × 7. LCM = 3 × 5 × 7 = 105.

Formula Used: LCM = highest powers of all prime factors

Q85 (Asked in 2015):

Find the HCF of 192 and 240 using Euclid’s algorithm.

Solution: Step 1: 240 = 192 × 1 + 48. Step 2: 192 = 48 × 4 + 0. HCF = 48.

Formula Used: a = bq + r

Q86 (Asked in 2014):

Express 900 as a product of its prime factors.

Solution: Divide 900: 900 ÷ 2 = 450, 450 ÷ 2 = 225, 225 ÷ 3 = 75, 75 ÷ 3 = 25, 25 ÷ 5 = 5, 5 ÷ 5 = 1. Thus, 900 = 2² × 3² × 5².

Formula Used: Fundamental Theorem of Arithmetic

Q87 (Asked in 2013):

Prove that √37 is irrational.

Solution: Assume √37 = p/q, where p, q are coprime, q ≠ 0. Then, 37 = p²/q², so p² = 37q². Thus, p² is divisible by 37, so p is divisible by 37 (p = 37m). Then, 1369m² = 37q², so q² = 37m². Thus, q is divisible by 37, contradicting coprimality. Hence, √37 is irrational.

Formula Used: Irrational numbers cannot be expressed as p/q

Q88 (Asked in 2012):

Find the LCM of 48 and 60.

Solution: Prime factorization: 48 = 2⁴ × 3, 60 = 2² × 3 × 5. LCM = 2⁴ × 3 × 5 = 240.

Formula Used: LCM = highest powers of all prime factors

Q89 (Asked in 2011):

Find the HCF of 147 and 343 using Euclid’s algorithm.

Solution: Step 1: 343 = 147 × 2 + 49. Step 2: 147 = 49 × 3 + 0. HCF = 49.

Formula Used: a = bq + r

Q90 (Asked in 2010):

Express 1728 as a product of its prime factors.

Solution: Divide 1728: 1728 ÷ 2 = 864, 864 ÷ 2 = 432, 432 ÷ 2 = 216, 216 ÷ 2 = 108, 108 ÷ 2 = 54, 54 ÷ 2 = 27, 27 ÷ 3 = 9, 9 ÷ 3 = 3, 3 ÷ 3 = 1. Thus, 1728 = 2⁶ × 3³.

Formula Used: Fundamental Theorem of Arithmetic

Q91 (Asked in 2024):

Show that 10 – √3 is irrational.

Solution: Assume 10 – √3 = p/q, where p, q are coprime, q ≠ 0. Then, √3 = 10 – p/q = (10q – p)/q. Since (10q – p)/q is rational and √3 is irrational, a contradiction arises. Hence, 10 – √3 is irrational.

Formula Used: Difference of rational and irrational is irrational

Q92 (Asked in 2023):

Find the LCM of 42 and 70.

Solution: Prime factorization: 42 = 2 × 3 × 7, 70 = 2 × 5 × 7. LCM = 2 × 3 × 5 × 7 = 210.

Formula Used: LCM = highest powers of all prime factors

Q93 (Asked in 2022):

Find the HCF of 216 and 288 using Euclid’s algorithm.

Solution: Step 1: 288 = 216 × 1 + 72. Step 2: 216 = 72 × 3 + 0. HCF = 72.

Formula Used: a = bq + r

Q94 (Asked in 2021):

Express 625 as a product of its prime factors.

Solution: Divide 625: 625 ÷ 5 = 125, 125 ÷ 5 = 25, 25 ÷ 5 = 5, 5 ÷ 5 = 1. Thus, 625 = 5⁴.

Formula Used: Fundamental Theorem of Arithmetic

Q95 (Asked in 2020):

Prove that √41 is irrational.

Solution: Assume √41 = p/q, where p, q are cop

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(M.Sc Maths, B. Sc, B.Ed, TGT Qualified 2016, UPTET Qualified)