📘 Quiz

Solution: Step 1: For divisibility by 72, the number must be divisible by both 8 and 9. Step 2: Check divisibility by 8: The last three digits '22K' must be divisible by 8. Step 3: Test values for K: Only '224' is divisible by 8, so K = 4. Step 4: Verify divisibility by 9: Sum of digits 4+3+3+2+2+4 = 18 is divisible by 9. Step 5: Therefore, K = 4.

Solution: Step 1: Factorize 4725 into prime factors: 4725 = 3^3 * 5^2 * 7 Step 2: Determine the powers of 3, 5, and 7 in the product of even numbers from 2 to 100. Step 3: For 7: There are 7, 14, 21... 98. So, 7 appears 8 times (considering 98 = 2 * 7^2). Step 4: For 5: 5, 10, 15... 100 contribute 5^12. Step 5: For 3: 3, 6, 9... 99 contribute more than 3^16. Step 6: Since 4725 = 3^3 * 5^2 * 7, the limiting factor is 7^8 and 5^12, and 3^3. Step 7: Hence, the largest 'm' is determined by the smallest quotient which is 6 for 5^12.

Solution: Step 1: Recall the divisibility rule for 3: A number is divisible by 3 if the sum of its digits is divisible by 3. Step 2: Calculate the sum of known digits: 2 + 3 + 4 + 9 + 8 = 26. Step 3: Determine the condition for divisibility by 3: 26 + X must be a multiple of 3. Step 4: Find the smallest multiple of 3 greater than 26: 27. Step 5: Calculate the minimum value of X: 27 - 26 = 1. Step 6: Find the largest multiple of 3 that is reasonably achievable with a single digit: 33. Step 7: Calculate the maximum value of X: 33 - 26 = 7. Step 8: Calculate the sum of the maximum and minimum possible values of X: 1 + 7 = 8.

Solution: Step 1: Identify the conditions for divisibility by 7, 12, 17, and 19 Step 2: Find the least common multiple (LCM) of 7 and 12 Step 3: List multiples of the LCM that are three-digit numbers Step 4: Check which multiple satisfies the remainder conditions for 17 and 19 Step 5: Determine the number to be added to make it a perfect square