Find the largest number that divides 400, 500, and 600,

Solution

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.