Set A Solutions

A1. Answer: (b) 97 Check each: 91 = 7×13 (composite); 97 has no factor other than 1 and itself (not divisible by 2,3,5,7 — and 11²=121>97, so checking primes up to √97≈9.8 suffices) → prime; 111 = 3×37 (composite); 121 = 11² (composite). Only 97 is prime.

A2. Answer: (d) 6000 In 46,281 the digits from left are 4(ten-thousands), 6(thousands), 2(hundreds), 8(tens), 1(units). The digit 6 is in the thousands place → place value = 6 × 1000 = 6000.

A3. Answer: (c) 210 Sum of first n natural numbers = n(n+1)/2 = 20×21/2 = 210.

A4. Answer: (b) 8, 9 15=3×5, 20=2²×5 → common factor 5, not co-prime. 8=2³, 9=3² → no common factor, HCF=1 → co-prime. 12=2²×3, 18=2×3² → common factors 2,3, not co-prime. 14=2×7, 21=3×7 → common factor 7, not co-prime.

A5. Answer: (a) 7 Unit digit of base 7 → cycle {7,9,3,1}, length 4. 45 ÷ 4 = 11 remainder 1 → 1st term of cycle = 7.

A6. Answer: (b) 6 12 = 2²×3, 18 = 2×3². HCF = 2¹×3¹ = 6.

A7. Answer: (c) 12 4 = 2², 6 = 2×3. LCM = 2²×3 = 12.

A8. Answer: (b) 5148 Alternating sum (right to left) for 5148: digits 8,4,1,5 → (8+1) − (4+5) = 9−9 = 0 → divisible by 11. Checking others: 4567 → (7+5)-(6+4)=12-10=2, not divisible; 6789 → (9+7)-(8+6)=16-14=2, not divisible; 8912 → (2+9)-(1+8)=11-9=2, not divisible.

A9. Answer: (c) 9 36 = 2² × 3². Number of factors = (2+1)(2+1) = 9.

A10. Answer: (c) 12 Product of the two numbers = HCF × LCM = 4 × 48 = 192. Other number = 192 ÷ 16 = 12.

A11. Answer: (c) √7 √16=4 (rational), √25=5 (rational), √7 is irrational (7 is not a perfect square), 0.333… = 1/3 (rational, recurring decimal).

A12. Answer: (c) 5 29 = 6×4 + 5 → remainder = 5.

A13. Answer: (a) 144 144 ÷ 12 = 12 exact. Check via 3 and 4: digit sum of 144=9 (div by 3), last two digits 44 ÷4=11 (div by 4) → divisible by 12. 128: digit sum 11, not div by 3 → fails. 162: last two digits 62÷4 not exact → fails. 134: digit sum 8, not div by 3 → fails.

A14. Answer: (c) 2 0 and 1 are not prime by definition; 2 is the smallest prime (and the only even prime); 3 is prime but not the smallest.

A15. Answer: (b) 16 2⁵ × 2³ ÷ 2⁴ = 2^(5+3−4) = 2⁴ = 16.

A16. Answer: (c) 5 (101)₂ = 1×2² + 0×2¹ + 1×2⁰ = 4 + 0 + 1 = 5.

A17. Answer: (a) 7√2 √98 = √(49 × 2) = 7√2 (49 is the largest perfect square factor of 98).

A18. Answer: (c) 7245 Digit sums: 4562 → 4+5+6+2=17 (not divisible by 9); 6531 → 6+5+3+1=15 (not divisible by 9); 7245 → 7+2+4+5=18 (divisible by 9); 8321 → 8+3+2+1=14 (not divisible by 9). Only 7245 qualifies.

A19. Answer: (c) 42 Pattern is n(n+1): 1×2=2, 2×3=6, 3×4=12, 4×5=20, 5×6=30, 6×7=42. Next term = 42.

A20. Answer: (c) 9 100 = 2² × 5². Number of factors = (2+1)(2+1) = 9.