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08 Jan 2025, 03:31
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How many 5-digit positive integers, where each digit is a prime number, are divisible by 25?
A. 64
B. 96
C. 128
D. 192
E. 256
A. 64
B. 96
C. 128
D. 192
E. 256
08 Jan 2025, 03:31
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Official Solution:
How many 5-digit positive integers, where each digit is a prime number, are divisible by 25?
A. 64
B. 96
C. 128
D. 192
E. 256
For a number to be divisible by 25, it must end in 00, 25, 50, or 75. Since we are looking for numbers composed entirely of prime digits, only 25 and 75 are valid options for the last two digits. Thus, there are two possibilities for the last two digits.
The prime digits are 2, 3, 5, and 7, meaning each of the first three digits can independently take any of these four values. This gives a total of 4 * 4 * 4 = 64 combinations for the first three digits.
Therefore, the total number of five-digit positive integers, where each digit is a prime number and the number is divisible by 25, is 2 * 64 = 128.
Answer: C
How many 5-digit positive integers, where each digit is a prime number, are divisible by 25?
A. 64
B. 96
C. 128
D. 192
E. 256
For a number to be divisible by 25, it must end in 00, 25, 50, or 75. Since we are looking for numbers composed entirely of prime digits, only 25 and 75 are valid options for the last two digits. Thus, there are two possibilities for the last two digits.
The prime digits are 2, 3, 5, and 7, meaning each of the first three digits can independently take any of these four values. This gives a total of 4 * 4 * 4 = 64 combinations for the first three digits.
Therefore, the total number of five-digit positive integers, where each digit is a prime number and the number is divisible by 25, is 2 * 64 = 128.
Answer: C
04 May 2026, 00:46
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How many 5-digit positive integers, where each digit is a prime number, are divisible by 25?
Count the prime digits:
2, 3, 5, 7
Number of Prime Digits = 4
Start with five slots for the five digits in the integers:
_ _ _ _ _
Fill each slot with the number of different digits that can be in that slot:
All numbers divisible by 5 end in 5 or 0. Of those, we have only the digit 5 available. So, the last digit can be only 5. There is just 1 possible last digit
_ _ _ _ 1
For the next digit only 2 and 7 work since 25 and 75 are divisible by 25 but not 35 or 55. There are 2 possibilities for the next digit.
_ _ _ 2 1
For the other three digits, any of the 4 primes will work since all multiples of 100, 1000, or 10,000 are divisible by 25.
4 4 4 2 1
Multiply the values in the five slots:
4 × 4 × 4 × 2 × 1 = 128
A. 64
B. 96
C. 128
D. 192
E. 256
Correct answer: C
Count the prime digits:
2, 3, 5, 7
Number of Prime Digits = 4
Start with five slots for the five digits in the integers:
_ _ _ _ _
Fill each slot with the number of different digits that can be in that slot:
All numbers divisible by 5 end in 5 or 0. Of those, we have only the digit 5 available. So, the last digit can be only 5. There is just 1 possible last digit
_ _ _ _ 1
For the next digit only 2 and 7 work since 25 and 75 are divisible by 25 but not 35 or 55. There are 2 possibilities for the next digit.
_ _ _ 2 1
For the other three digits, any of the 4 primes will work since all multiples of 100, 1000, or 10,000 are divisible by 25.
4 4 4 2 1
Multiply the values in the five slots:
4 × 4 × 4 × 2 × 1 = 128
A. 64
B. 96
C. 128
D. 192
E. 256
Correct answer: C
