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12 Jan 2026, 11:27
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What is the sum of all five digit positive integers that can be formed using each of the digits 0, 2, 3, 5, and 7 exactly once?
A. 113,322
B. 4,419,660
C. 4,419,964
D. 4,419,966
E. 4,533,288
A. 113,322
B. 4,419,660
C. 4,419,964
D. 4,419,966
E. 4,533,288
12 Jan 2026, 11:27
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Official Solution:
What is the sum of all five digit positive integers that can be formed using each of the digits 0, 2, 3, 5, and 7 exactly once?
A. 113,322
B. 4,419,660
C. 4,419,964
D. 4,419,966
E. 4,533,288
Since the numbers in the options are quite large, calculating the entire sum directly would take a long time. Therefore, we should look for a shortcut. Notice that each option has a different unit digit, so focusing on the unit digit is a good strategy.
First, let’s find the unit digit of the sum if 0 were allowed as the first digit. Using the digits 0, 2, 3, 5, and 7 exactly once, there are 5! arrangements. In these arrangements, each digit appears equally often in the units place, specifically 5!/5 = 24 times. So, the unit digit of the total sum is the unit digit of:
\(24 * (0 + 2 + 3 + 5 + 7) = 24 * 17\)
The unit digit of 24 * 17 is 8. Therefore, the unit digit of this larger sum is 8.
Next, let’s find the unit digit of the sum of the numbers where 0 is fixed as the first digit. With 0 as the first digit, the remaining digits (2, 3, 5, 7) form 4! arrangements. Each of these four digits appears 4!/4 = 6 times in the units place. So, the unit digit of the sum of these numbers is the unit digit of:
\(6 * (2 + 3 + 5 + 7) = 6 * 17\)
The unit digit of 6 * 17 is 2. Therefore, the unit digit of this excluded sum is 2.
Finally, the unit digit of the required sum is:
\(8 - 2 = 6\).
To clarify, we are first calculating the unit digit of the sum assuming 0 is allowed as the first digit, and then subtracting the unit digit of the sum where 0 is fixed as the first digit. This gives us the unit digit of the sum where 0 is not the first digit, which is the unit digit we need for the total sum of all the five-digit numbers.
Only one option has 6 as the unit digit, so that option must be correct.
Answer: D
What is the sum of all five digit positive integers that can be formed using each of the digits 0, 2, 3, 5, and 7 exactly once?
A. 113,322
B. 4,419,660
C. 4,419,964
D. 4,419,966
E. 4,533,288
Since the numbers in the options are quite large, calculating the entire sum directly would take a long time. Therefore, we should look for a shortcut. Notice that each option has a different unit digit, so focusing on the unit digit is a good strategy.
First, let’s find the unit digit of the sum if 0 were allowed as the first digit. Using the digits 0, 2, 3, 5, and 7 exactly once, there are 5! arrangements. In these arrangements, each digit appears equally often in the units place, specifically 5!/5 = 24 times. So, the unit digit of the total sum is the unit digit of:
\(24 * (0 + 2 + 3 + 5 + 7) = 24 * 17\)
The unit digit of 24 * 17 is 8. Therefore, the unit digit of this larger sum is 8.
Next, let’s find the unit digit of the sum of the numbers where 0 is fixed as the first digit. With 0 as the first digit, the remaining digits (2, 3, 5, 7) form 4! arrangements. Each of these four digits appears 4!/4 = 6 times in the units place. So, the unit digit of the sum of these numbers is the unit digit of:
\(6 * (2 + 3 + 5 + 7) = 6 * 17\)
The unit digit of 6 * 17 is 2. Therefore, the unit digit of this excluded sum is 2.
Finally, the unit digit of the required sum is:
\(8 - 2 = 6\).
To clarify, we are first calculating the unit digit of the sum assuming 0 is allowed as the first digit, and then subtracting the unit digit of the sum where 0 is fixed as the first digit. This gives us the unit digit of the sum where 0 is not the first digit, which is the unit digit we need for the total sum of all the five-digit numbers.
Only one option has 6 as the unit digit, so that option must be correct.
Answer: D
liberoasperiores
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25 Jan 2026, 03:24
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Awesome Question !
Bunuel
