Playing with Numbers

11:00 AM

Step 1: The bells will ring together again after the LCM of their intervals. Step 2: Find LCM of 12, 18, 20. Prime factorisations: 12=2²×3, 18=2×3², 20=2²×5. LCM = 2²×3²×5 = 4×9×5 = 180 minutes. Step 3: Convert 180 minutes to hours: 180÷60 = 3 hours. Step 4: Add 3 hours to 8:00 AM: 8:00 AM + 3 hours = 11:00 AM. The bells will ring together again at 11:00 AM.

None of these — see explanation

Step 1: Divisibility rule for 11: Find difference between sum of digits at odd places (from right) and sum of digits at even places. If difference is 0 or divisible by 11, the number is divisible by 11. Step 2: For 935142 — odd places (right): 2,1,3 = 6; even places: 4,5,9 = 18; difference = 18−6 = 12. Not divisible by 11. Step 3: For 856743 — odd places: 3,7,5 = 15; even: 4,6,8 = 18; diff = 3. Not divisible by 11. Step 4: This question tests the application of divisibility rule for 11. Always subtract smaller from larger and check if result is 0 or a multiple of 11.

1080

Step 1: The number must be exactly divisible by 6, 8, and 9. So it must be a multiple of LCM(6, 8, 9). Step 2: Prime factorisations: 6=2×3, 8=2³, 9=3². LCM = 2³×3² = 8×9 = 72. Step 3: We need the smallest 4-digit multiple of 72. The smallest 4-digit number is 1000. Divide: 1000÷72 = 13.88... So next complete multiple = 14×72 = 1008. Step 4: Check: 1008÷6=168 ✓, 1008÷8=126 ✓, 1008÷9=112 ✓. Wait — 14×72=1008 ✓. So the answer is 1008. Note: 1080=15×72 is also valid but 1008 is the SMALLEST. The correct answer is 1008.

9696

Step 1: Use the formula: HCF × LCM = Product of two numbers. Step 2: Product of 96 and 404 = 96 × 404. Let us calculate: 96×400 = 38400, 96×4 = 384. So 96×404 = 38400+384 = 38784. Step 3: LCM = Product ÷ HCF = 38784 ÷ 4 = 9696. Step 4: Verify by prime factorisation: 96=2⁵×3, 404=2²×101. HCF=2²=4 ✓. LCM=2⁵×3×101=32×303=9696 ✓. The answer is 9696.