{
  "title": "Fundamental Theorem of Arithmetic - Grade 10 CBSE",
  "total_questions": 100,
  "questions": [
    {"id": 1, "difficulty": "easy", "question": "Fundamental Theorem of Arithmetic states that every composite number can be expressed as:", "options": {"A": "Product of primes uniquely", "B": "Sum of primes", "C": "Difference of primes", "D": "Power of primes"}, "correct_answer": "A"},
    {"id": 2, "difficulty": "easy", "question": "Prime factorization of 12 is:", "options": {"A": "2×6", "B": "3×4", "C": "2²×3", "D": "2×2×2×3"}, "correct_answer": "C"},
    {"id": 3, "difficulty": "easy", "question": "Express 20 as product of primes.", "options": {"A": "2×10", "B": "4×5", "C": "2²×5", "D": "2×2×3"}, "correct_answer": "C"},
    {"id": 4, "difficulty": "easy", "question": "Prime factors of 28 are:", "options": {"A": "2,7", "B": "2,14", "C": "4,7", "D": "1,28"}, "correct_answer": "A"},
    {"id": 5, "difficulty": "easy", "question": "Express 30 as product of primes.", "options": {"A": "2×15", "B": "3×10", "C": "2×3×5", "D": "5×6"}, "correct_answer": "C"},
    {"id": 6, "difficulty": "easy", "question": "Prime factorization of 42:", "options": {"A": "2×21", "B": "3×14", "C": "2×3×7", "D": "6×7"}, "correct_answer": "C"},
    {"id": 7, "difficulty": "easy", "question": "Express 45 as product of primes.", "options": {"A": "3×15", "B": "5×9", "C": "3²×5", "D": "3×3×3×5"}, "correct_answer": "C"},
    {"id": 8, "difficulty": "easy", "question": "Prime factors of 60:", "options": {"A": "2,3,5", "B": "2,30", "C": "3,20", "D": "4,15"}, "correct_answer": "A"},
    {"id": 9, "difficulty": "easy", "question": "Express 72 as product of primes.", "options": {"A": "8×9", "B": "2³×3²", "C": "2×36", "D": "3×24"}, "correct_answer": "B"},
    {"id": 10, "difficulty": "easy", "question": "Prime factorization of 100:", "options": {"A": "10×10", "B": "2²×5²", "C": "4×25", "D": "2×50"}, "correct_answer": "B"},
    {"id": 11, "difficulty": "easy", "question": "Express 84 as product of primes.", "options": {"A": "2×42", "B": "3×28", "C": "2²×3×7", "D": "4×21"}, "correct_answer": "C"},
    {"id": 12, "difficulty": "easy", "question": "Prime factors of 210:", "options": {"A": "2,3,5,7", "B": "2,105", "C": "3,70", "D": "5,42"}, "correct_answer": "A"},
    {"id": 13, "difficulty": "easy", "question": "Express 120 as product of primes.", "options": {"A": "2³×3×5", "B": "2×60", "C": "3×40", "D": "4×30"}, "correct_answer": "A"},
    {"id": 14, "difficulty": "easy", "question": "Prime factorization of 144:", "options": {"A": "12×12", "B": "2⁴×3²", "C": "2×72", "D": "3×48"}, "correct_answer": "B"},
    {"id": 15, "difficulty": "easy", "question": "Express 180 as product of primes.", "options": {"A": "2²×3²×5", "B": "2×90", "C": "3×60", "D": "4×45"}, "correct_answer": "A"},
    {"id": 16, "difficulty": "easy", "question": "Prime factors of 250:", "options": {"A": "2×5³", "B": "2×125", "C": "5×50", "D": "10×25"}, "correct_answer": "A"},
    {"id": 17, "difficulty": "easy", "question": "Express 300 as product of primes.", "options": {"A": "2²×3×5²", "B": "3×100", "C": "4×75", "D": "6×50"}, "correct_answer": "A"},
    {"id": 18, "difficulty": "easy", "question": "Prime factorization of 360:", "options": {"A": "2³×3²×5", "B": "2×180", "C": "3×120", "D": "6×60"}, "correct_answer": "A"},
    {"id": 19, "difficulty": "easy", "question": "Express 400 as product of primes.", "options": {"A": "2⁴×5²", "B": "4×100", "C": "8×50", "D": "16×25"}, "correct_answer": "A"},
    {"id": 20, "difficulty": "easy", "question": "Prime factors of 500:", "options": {"A": "2²×5³", "B": "2×250", "C": "4×125", "D": "5×100"}, "correct_answer": "A"},
    
    {"id": 21, "difficulty": "medium", "question": "Find LCM of 12 and 18 using prime factorization.", "options": {"A": "24", "B": "36", "C": "48", "D": "54"}, "correct_answer": "B"},
    {"id": 22, "difficulty": "medium", "question": "Find HCF of 36 and 48 using prime factorization.", "options": {"A": "6", "B": "12", "C": "18", "D": "24"}, "correct_answer": "B"},
    {"id": 23, "difficulty": "medium", "question": "Find LCM of 24 and 36 using prime factors.", "options": {"A": "48", "B": "60", "C": "72", "D": "84"}, "correct_answer": "C"},
    {"id": 24, "difficulty": "medium", "question": "Find HCF of 56 and 98 using prime factorization.", "options": {"A": "7", "B": "14", "C": "28", "D": "42"}, "correct_answer": "B"},
    {"id": 25, "difficulty": "medium", "question": "Find LCM of 15 and 25 using prime factors.", "options": {"A": "50", "B": "75", "C": "100", "D": "125"}, "correct_answer": "B"},
    {"id": 26, "difficulty": "medium", "question": "Find HCF of 75 and 125 using prime factorization.", "options": {"A": "5", "B": "15", "C": "25", "D": "50"}, "correct_answer": "C"},
    {"id": 27, "difficulty": "medium", "question": "Find LCM of 28 and 42 using prime factors.", "options": {"A": "56", "B": "84", "C": "112", "D": "126"}, "correct_answer": "B"},
    {"id": 28, "difficulty": "medium", "question": "Find HCF of 84 and 108 using prime factorization.", "options": {"A": "6", "B": "12", "C": "18", "D": "24"}, "correct_answer": "B"},
    {"id": 29, "difficulty": "medium", "question": "Find LCM of 40 and 60 using prime factors.", "options": {"A": "80", "B": "100", "C": "120", "D": "140"}, "correct_answer": "C"},
    {"id": 30, "difficulty": "medium", "question": "Find HCF of 96 and 144 using prime factorization.", "options": {"A": "12", "B": "24", "C": "36", "D": "48"}, "correct_answer": "D"},
    {"id": 31, "difficulty": "medium", "question": "Find LCM of 18 and 24 using prime factorization.", "options": {"A": "36", "B": "48", "C": "60", "D": "72"}, "correct_answer": "D"},
    {"id": 32, "difficulty": "medium", "question": "Find HCF of 120 and 150 using prime factors.", "options": {"A": "10", "B": "15", "C": "30", "D": "45"}, "correct_answer": "C"},
    {"id": 33, "difficulty": "medium", "question": "Find LCM of 45 and 75 using prime factorization.", "options": {"A": "150", "B": "175", "C": "225", "D": "275"}, "correct_answer": "C"},
    {"id": 34, "difficulty": "medium", "question": "Find HCF of 210 and 350 using prime factors.", "options": {"A": "35", "B": "70", "C": "105", "D": "140"}, "correct_answer": "B"},
    {"id": 35, "difficulty": "medium", "question": "Find LCM of 16 and 20 using prime factorization.", "options": {"A": "40", "B": "60", "C": "80", "D": "100"}, "correct_answer": "C"},
    {"id": 36, "difficulty": "medium", "question": "Find HCF of 180 and 324 using prime factors.", "options": {"A": "18", "B": "36", "C": "54", "D": "72"}, "correct_answer": "B"},
    {"id": 37, "difficulty": "medium", "question": "Find LCM of 54 and 72 using prime factorization.", "options": {"A": "144", "B": "162", "C": "216", "D": "288"}, "correct_answer": "C"},
    {"id": 38, "difficulty": "medium", "question": "Find HCF of 280 and 420 using prime factors.", "options": {"A": "35", "B": "70", "C": "105", "D": "140"}, "correct_answer": "D"},
    {"id": 39, "difficulty": "medium", "question": "Find LCM of 63 and 84 using prime factorization.", "options": {"A": "126", "B": "168", "C": "252", "D": "336"}, "correct_answer": "C"},
    {"id": 40, "difficulty": "medium", "question": "Find HCF of 385 and 525 using prime factors.", "options": {"A": "35", "B": "70", "C": "105", "D": "175"}, "correct_answer": "A"},
    {"id": 41, "difficulty": "medium", "question": "Find LCM of 48 and 64 using prime factorization.", "options": {"A": "96", "B": "128", "C": "192", "D": "256"}, "correct_answer": "C"},
    {"id": 42, "difficulty": "medium", "question": "Find HCF of 432 and 504 using prime factors.", "options": {"A": "36", "B": "48", "C": "54", "D": "72"}, "correct_answer": "D"},
    {"id": 43, "difficulty": "medium", "question": "Find LCM of 75 and 90 using prime factorization.", "options": {"A": "150", "B": "225", "C": "300", "D": "450"}, "correct_answer": "D"},
    {"id": 44, "difficulty": "medium", "question": "Find HCF of 616 and 1300 using prime factors.", "options": {"A": "2", "B": "4", "C": "8", "D": "16"}, "correct_answer": "B"},
    {"id": 45, "difficulty": "medium", "question": "Find LCM of 96 and 120 using prime factorization.", "options": {"A": "240", "B": "360", "C": "480", "D": "600"}, "correct_answer": "C"},
    {"id": 46, "difficulty": "medium", "question": "Find HCF of 693 and 1071 using prime factors.", "options": {"A": "21", "B": "42", "C": "63", "D": "84"}, "correct_answer": "C"},
    {"id": 47, "difficulty": "medium", "question": "Find LCM of 140 and 196 using prime factorization.", "options": {"A": "280", "B": "392", "C": "490", "D": "980"}, "correct_answer": "D"},
    {"id": 48, "difficulty": "medium", "question": "Find HCF of 850 and 680 using prime factors.", "options": {"A": "85", "B": "170", "C": "255", "D": "340"}, "correct_answer": "B"},
    {"id": 49, "difficulty": "medium", "question": "Find LCM of 150 and 225 using prime factorization.", "options": {"A": "300", "B": "375", "C": "450", "D": "525"}, "correct_answer": "C"},
    {"id": 50, "difficulty": "medium", "question": "Find HCF of 1260 and 1440 using prime factors.", "options": {"A": "60", "B": "90", "C": "120", "D": "180"}, "correct_answer": "D"},
    {"id": 51, "difficulty": "medium", "question": "Find LCM of 210 and 350 using prime factorization.", "options": {"A": "700", "B": "1050", "C": "1400", "D": "1750"}, "correct_answer": "B"},
    {"id": 52, "difficulty": "medium", "question": "Find HCF of 272 and 425 using prime factors.", "options": {"A": "17", "B": "34", "C": "51", "D": "68"}, "correct_answer": "A"},
    {"id": 53, "difficulty": "medium", "question": "Find LCM of 360 and 504 using prime factorization.", "options": {"A": "720", "B": "1080", "C": "1440", "D": "2520"}, "correct_answer": "D"},
    {"id": 54, "difficulty": "medium", "question": "Find HCF of 441, 567 and 693 using prime factors.", "options": {"A": "21", "B": "42", "C": "63", "D": "84"}, "correct_answer": "C"},
    {"id": 55, "difficulty": "medium", "question": "Find LCM of 240 and 330 using prime factorization.", "options": {"A": "1320", "B": "1980", "C": "2640", "D": "3300"}, "correct_answer": "C"},
    {"id": 56, "difficulty": "medium", "question": "Find HCF of 588 and 980 using prime factors.", "options": {"A": "28", "B": "56", "C": "84", "D": "112"}, "correct_answer": "A"},
    {"id": 57, "difficulty": "medium", "question": "Find LCM of 385 and 525 using prime factorization.", "options": {"A": "1155", "B": "2310", "C": "3465", "D": "5775"}, "correct_answer": "D"},
    {"id": 58, "difficulty": "medium", "question": "Find HCF of 1260, 1440 and 1620 using prime factors.", "options": {"A": "60", "B": "90", "C": "120", "D": "180"}, "correct_answer": "D"},
    {"id": 59, "difficulty": "medium", "question": "Find LCM of 432 and 504 using prime factorization.", "options": {"A": "864", "B": "1296", "C": "1512", "D": "3024"}, "correct_answer": "D"},
    {"id": 60, "difficulty": "medium", "question": "Find HCF of 360, 504 and 672 using prime factors.", "options": {"A": "12", "B": "24", "C": "36", "D": "48"}, "correct_answer": "B"},
    {"id": 61, "difficulty": "medium", "question": "Check if 6^n can end with digit 0 for any natural n.", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Only for n=1"}, "correct_answer": "B"},
    {"id": 62, "difficulty": "medium", "question": "Check if 4^n can end with digit 0 for any natural n.", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Only for n=5"}, "correct_answer": "B"},
    {"id": 63, "difficulty": "medium", "question": "Can 12^n end with digit 0? (n natural)", "options": {"A": "Yes", "B": "No", "C": "Only if n=5", "D": "Only if n=10"}, "correct_answer": "B"},
    {"id": 64, "difficulty": "medium", "question": "Check if 15^n can end with digit 5.", "options": {"A": "Yes, always", "B": "No", "C": "Only odd n", "D": "Only even n"}, "correct_answer": "A"},
    {"id": 65, "difficulty": "medium", "question": "Can 8^n end with digit 2?", "options": {"A": "Yes", "B": "No", "C": "Only for n=1", "D": "Only for n=3"}, "correct_answer": "A"},
    {"id": 66, "difficulty": "medium", "question": "Check if 5^n can end with digit 2.", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Only for n=0"}, "correct_answer": "B"},
    {"id": 67, "difficulty": "medium", "question": "Can 3^n end with digit 1?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Only for n=0"}, "correct_answer": "A"},
    {"id": 68, "difficulty": "medium", "question": "Check if 7^n can end with digit 3.", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Only for n=1"}, "correct_answer": "A"},
    {"id": 69, "difficulty": "medium", "question": "Can 2^n end with digit 6?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Only for n=4"}, "correct_answer": "A"},
    {"id": 70, "difficulty": "medium", "question": "Check if 9^n can end with digit 4.", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Only for n=2"}, "correct_answer": "A"},
    {"id": 71, "difficulty": "medium", "question": "Can 10^n end with digit 0?", "options": {"A": "Yes, always", "B": "No", "C": "Only odd n", "D": "Only even n"}, "correct_answer": "A"},
    {"id": 72, "difficulty": "medium", "question": "Check if 11^n can end with digit 1.", "options": {"A": "Yes, always", "B": "No", "C": "Only odd n", "D": "Only even n"}, "correct_answer": "A"},
    {"id": 73, "difficulty": "medium", "question": "Can 14^n end with digit 6?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Only for n=1"}, "correct_answer": "A"},
    {"id": 74, "difficulty": "medium", "question": "Check if 20^n can end with digit 0.", "options": {"A": "Yes, always", "B": "No", "C": "Only odd n", "D": "Only even n"}, "correct_answer": "A"},
    {"id": 75, "difficulty": "medium", "question": "Can 25^n end with digit 5?", "options": {"A": "Yes, always", "B": "No", "C": "Only odd n", "D": "Only even n"}, "correct_answer": "A"},
    {"id": 76, "difficulty": "medium", "question": "Check if 30^n can end with digit 0.", "options": {"A": "Yes, always", "B": "No", "C": "Only odd n", "D": "Only even n"}, "correct_answer": "A"},
    {"id": 77, "difficulty": "medium", "question": "Can 16^n end with digit 2?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Only for n=1"}, "correct_answer": "B"},
    {"id": 78, "difficulty": "medium", "question": "Check if 18^n can end with digit 8.", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Only for n=1"}, "correct_answer": "A"},
    {"id": 79, "difficulty": "medium", "question": "Can 24^n end with digit 4?", "options": {"A": "Yes", "B": "No", "C": "Sometimes", "D": "Only for n=1"}, "correct_answer": "A"},
    {"id": 80, "difficulty": "medium", "question": "Check if 36^n can end with digit 6.", "options": {"A": "Yes, always", "B": "No", "C": "Only odd n", "D": "Only even n"}, "correct_answer": "A"},
    
    {"id": 81, "difficulty": "hard", "question": "If HCF(306, 657) = 9, find LCM(306, 657).", "options": {"A": "22338", "B": "22348", "C": "22358", "D": "22368"}, "correct_answer": "A"},
    {"id": 82, "difficulty": "hard", "question": "If LCM(26, 169) = 338, find HCF(26, 169).", "options": {"A": "13", "B": "26", "C": "39", "D": "52"}, "correct_answer": "A"},
    {"id": 83, "difficulty": "hard", "question": "Given HCF(96, 404) = 4, find LCM(96, 404).", "options": {"A": "9696", "B": "9796", "C": "9896", "D": "9996"}, "correct_answer": "A"},
    {"id": 84, "difficulty": "hard", "question": "If LCM(60, 72) = 360, find HCF(60, 72).", "options": {"A": "6", "B": "12", "C": "18", "D": "24"}, "correct_answer": "B"},
    {"id": 85, "difficulty": "hard", "question": "Given HCF(75, 125) = 25, find LCM(75, 125).", "options": {"A": "250", "B": "300", "C": "350", "D": "375"}, "correct_answer": "D"},
    {"id": 86, "difficulty": "hard", "question": "If LCM(84, 108) = 756, find HCF(84, 108).", "options": {"A": "6", "B": "12", "C": "18", "D": "24"}, "correct_answer": "B"},
    {"id": 87, "difficulty": "hard", "question": "Given HCF(210, 55) = 5, find LCM(210, 55).", "options": {"A": "2310", "B": "2320", "C": "2330", "D": "2340"}, "correct_answer": "A"},
    {"id": 88, "difficulty": "hard", "question": "If LCM(96, 144) = 288, find HCF(96, 144).", "options": {"A": "24", "B": "36", "C": "48", "D": "72"}, "correct_answer": "C"},
    {"id": 89, "difficulty": "hard", "question": "Given HCF(385, 525) = 35, find LCM(385, 525).", "options": {"A": "5775", "B": "5875", "C": "5975", "D": "6075"}, "correct_answer": "A"},
    {"id": 90, "difficulty": "hard", "question": "If LCM(150, 225) = 450, find HCF(150, 225).", "options": {"A": "25", "B": "50", "C": "75", "D": "100"}, "correct_answer": "C"},
    {"id": 91, "difficulty": "hard", "question": "Explain why 7×11×13+13 is composite.", "options": {"A": "It has factors other than 1 and itself", "B": "It's prime", "C": "It's even", "D": "It ends with 3"}, "correct_answer": "A"},
    {"id": 92, "difficulty": "hard", "question": "Explain why 5×7×11+7×3 is composite.", "options": {"A": "It has factors 7 and (5×11+3)", "B": "It's prime", "C": "It's odd", "D": "It ends with 5"}, "correct_answer": "A"},
    {"id": 93, "difficulty": "hard", "question": "Show that 6^n cannot end with digit 0 for any n∈N.", "options": {"A": "6=2×3, needs 5 as factor for ending 0", "B": "6 is even", "C": "6 is composite", "D": "6^n always even"}, "correct_answer": "A"},
    {"id": 94, "difficulty": "hard", "question": "Show that 12^n cannot end with digit 0 or 5.", "options": {"A": "12=2²×3, needs 5 as factor", "B": "12 is even", "C": "12 is composite", "D": "12^n always even"}, "correct_answer": "A"},
    {"id": 95, "difficulty": "hard", "question": "If n is odd, show that n²-1 is divisible by 8.", "options": {"A": "n=2k+1, then n²-1=4k(k+1)", "B": "n=3k+1", "C": "n=4k+1", "D": "n=5k+1"}, "correct_answer": "A"},
    {"id": 96, "difficulty": "hard", "question": "Show that square of any positive integer is of form 3m or 3m+1.", "options": {"A": "a=3q,3q+1,3q+2 then square", "B": "a=4q,4q+1", "C": "a=5q,5q+1", "D": "a=2q,2q+1"}, "correct_answer": "A"},
    {"id": 97, "difficulty": "hard", "question": "Show that cube of any positive integer is of form 9m, 9m+1 or 9m+8.", "options": {"A": "a=3q,3q+1,3q+2 then cube", "B": "a=4q,4q+1", "C": "a=5q,5q+1", "D": "a=2q,2q+1"}, "correct_answer": "A"},
    {"id": 98, "difficulty": "hard", "question": "For any positive integer n, prove n³-n is divisible by 6.", "options": {"A": "n³-n=(n-1)n(n+1), product of 3 consecutive", "B": "n³-n=n(n²-1)", "C": "n³-n=(n-1)(n+1)n", "D": "All of above"}, "correct_answer": "D"},
    {"id": 99, "difficulty": "hard", "question": "Show that √2 is irrational using Fundamental Theorem.", "options": {"A": "Assume √2=a/b in lowest terms, then 2=a²/b²", "B": "Assume √2 rational", "C": "Assume √2 integer", "D": "Assume √2 prime"}, "correct_answer": "A"},
    {"id": 100, "difficulty": "hard", "question": "Show that √3 is irrational using Fundamental Theorem.", "options": {"A": "Assume √3=a/b in lowest terms, then 3=a²/b²", "B": "Assume √3 rational", "C": "Assume √3 integer", "D": "Assume √3 prime"}, "correct_answer": "A"}
  ]
}