What is the greatest four-digit integer that meets the following three restrictions ?
All of the digits are different.
The greatest digit is the sum of the other three digits.
The product of the four digits is divisible by 10 and not equal to zero.
the lead digit should be 9
to be divisible by 10 , two of the digits must be 5 and 2 , or 5 and 4
We can rule out 5 and 6, and 5 and 8, since the sum would be greater than 9
let's try 952X, X must be 2, but all digits must be different
let's try 9540, no good, since we can't have a product of all the digits being zero
ok, how about the lead digit being 8
then we need 8521, yup, that's the one
To find the greatest four-digit integer that meets the given restrictions, let's consider the conditions one by one:
1. All of the digits are different:
Since the digits must be different, we need to find the largest digit for the greatest four-digit integer. The greatest digit is 9.
2. The greatest digit is the sum of the other three digits:
Let's assume the three digits are a, b, and c (in descending order). According to the given condition, 9 (the greatest digit) is equal to a + b + c.
Given that a, b, and c are all different, the maximum sum of three different digits can be obtained by taking the digits 7, 6, and 5. Therefore, a = 7, b = 6, and c = 5.
3. The product of the four digits is divisible by 10 and not equal to zero:
Since the product must be divisible by 10, we need to include the digit 0. All the digits are different, so the remaining digit must be 0. Thus, the digits are 9, 7, 6, and 0.
Combining all the digits in descending order, the greatest four-digit integer that meets the given restrictions is 9760.
To find the greatest four-digit integer that meets the given restrictions, we need to follow these steps:
Step 1: Determine the greatest digit
Since the greatest digit is the sum of the other three digits, we need to find three distinct digits whose sum is the greatest possible single digit. The greatest single digit is 9, so the sum of the other three digits should be 9.
Step 2: Determine the three remaining digits
To find three different digits whose sum is 9, we can start by assuming the largest digit as 9 and then testing different combinations. Based on trial and error, we find that the three remaining digits can be 8, 1, and 0.
Step 3: Check if the product of the four digits meets the conditions
The product of the four digits is divisible by 10 and not equal to zero. For this, we need to ensure that one of the digits is zero (to make it divisible by 10) and that none of the digits is zero (to satisfy the second condition). From step 2, we see that one of the digits is indeed zero, satisfying the condition.
Step 4: Arrange the digits to form the greatest four-digit integer
Now, we arrange the digits in descending order to form the greatest four-digit integer. Based on the digits we found in step 2, the greatest four-digit integer that satisfies all the conditions is 9801.
Therefore, the greatest four-digit integer that meets all the given restrictions is 9801.