For each of the three-digit numbers with no digits zero, the difference between the number itself and the product of its digits is calculated. The largest such difference is
A. 110
B. 270
C. 902
D. 910
E. 927
To find the largest difference between a three-digit number and the product of its digits, we need to consider all possible three-digit numbers with non-zero digits.
Let's start by finding the largest difference for a three-digit number in the form ABC, where A, B, and C are non-zero digits.
The number itself can be expressed as 100A + 10B + C, and the product of its digits is ABC = A * B * C.
So, the difference between the number and the product of its digits is:
(100A + 10B + C) - (A * B * C) = 100A + 10B + C - ABC = 100A + 10B + C - (A * B * C)
Now, since we want to find the largest difference, we need to maximize the value of (100A + 10B + C) - (A * B * C).
To maximize this difference, we need to choose the largest possible values for A, B, and C. Since the digits cannot be zero, the maximum value for each digit is 9.
Let's substitute 9 for A, B, and C in the above expression:
(100 * 9 + 10 * 9 + 9) - (9 * 9 * 9) = 909 - 729 = 180
Therefore, the largest difference for a three-digit number with non-zero digits is 180.
Among the answer choices, the largest difference is 180, so the answer is not listed.
To find the largest difference between a three-digit number and the product of its digits, we need to consider all possible three-digit numbers with no digits as zero.
Let's start by considering the smallest three-digit number, which is 123. The product of its digits is 1 * 2 * 3 = 6. The difference between 123 and 6 is 117.
Now, let's consider the largest three-digit number, which is 987. The product of its digits is 9 * 8 * 7 = 504. The difference between 987 and 504 is 483.
Next, let's consider 124. The product of its digits is 1 * 2 * 4 = 8. The difference between 124 and 8 is 116.
We can continue this process for all possible three-digit numbers with no digits as zero. However, there is a pattern we can observe.
The largest difference occurs when the three-digit number has its highest digit in the hundreds place and its lowest digit in the units place. Therefore, the largest difference occurs when the three-digit number is in the form of XYZ, where X > Y > Z.
Using this observation, the largest possible difference occurs when X = 9, Y = 8, and Z = 7, resulting in the number 987. We have already calculated the difference for this number, which is 987 - 504 = 483.
Therefore, the largest difference between a three-digit number and the product of its digits is 483, which is closest to option D.