Find the largest number that divides 400, 500, and 600, leaving remainders 10, 20, and 30 respectively.

Let the required number be x. When dividing 400 by x, the remainder is 10, so 400 - 10 = 390 is divisible by x. When dividing 500 by x, the remainder is 20, so 500 - 20 = 480 is divisible by x. When dividing 600 by x, the remainder is 30, so 600 - 30 = 570 is divisible by x. Now, we need to find the greatest common divisor (GCD) of 390, 480, and 570. Prime factorizations: 390 = 2 * 3 * 5 * 13 480 = 2^5 * 3 * 5 570 = 2 * 3 * 5 * 19 The GCD is 2 * 3 * 5 = 30. Therefore, the required number is 30.