Number System

The HCF of a set of numbers must be a divisor of their LCM. 150 is divisible by 15, 25, and 50, but not by 55.

Number divides 690-10=680 and 875-25=850. GCD(680, 850) = 170. The greatest number is 170. Product of digits = 1 * 7 * 0 = 0.

Numbers with HCF 531 must be multiples of 531. We check multiples of 531 near 9999: 531 * 19 = 10089 (too large), 531 * 18 = 9558, and 531 * 17 = 9027.

For two numbers with HCF 45 and LCM 540, their product is 45 * 540 = 24300. The numbers must be multiples of 45, such as 45a and 45b where a and b are coprime. 45a * 45b = 24300 implies ab = 12. Possible pairs for (a,b) are (1,12) and (3,4). The numbers are (45, 540) or (135, 180). Since both must have the same number of digits, 135 and 180 are the correct pair.

The differences between divisors and remainders are constant: 4-1=3, 6-3=3, 14-11=3, 20-17=3. Find the LCM of (4, 6, 14, 20) = 420. The largest 5-digit number is 99999. 99999 / 420 = 238 with remainder 39. The largest multiple is 99999 - 39 = 99960. Subtract the common difference 3 to get 99957.

Let the numbers be 29a and 29b where a and b are coprime. 29 * a * b = 4147, so ab = 143. Factors of 143 are (1, 143) and (11, 13). Since both numbers > 29, we use 11 and 13. The numbers are 29*11 = 319 and 29*13 = 377. Sum = 319 + 377 = 696.

To find the greatest number, subtract the remainders from the numbers: 445-4=441, 572-5=567, and 699-6=693. The greatest common divisor of 441, 567, and 693 is 63.

Smallest odd composite number is 9. Largest three-digit prime is 997. HCF(9, 997) = 1 (since 997 is prime and not divisible by 3). LCM(9, 997) = 9 * 997 = 8973. Difference = 8973 - 1 = 8972.

Let divisor be D and quotient be Q. D = Q. Dividend = D * Q + R = D^2 + R. Given R = D/2. So 6123 = D^2 + D/2. Multiply by 2: 12246 = 2D^2 + D. 2D^2 + D - 12246 = 0. Using quadratic formula: D = (-1 + sqrt(1 + 4*2*12246)) / 4 = (-1 + sqrt(97969)) / 4 = (-1 + 313) / 4 = 312 / 4 = 78.