How many five-digit numbers can be formed from the digits 0, 1, 2, 3, 4, and 5, if no digits can repeat and the number must be divisible by 4?
A. 36
B. 48
C. 72
D. 96
E. 144
The OA is the option E.
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How many five-digit numbers can be formed from the digits
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For the 5-digit integer to be a multiple of 4, the last two digits must form a multiple of 4.VJesus12 wrote:How many five-digit numbers can be formed from the digits 0, 1, 2, 3, 4, and 5, if no digits can repeat and the number must be divisible by 4?
A. 36
B. 48
C. 72
D. 96
E. 144
Options:
04, 12, 20, 24, 32, 40, 52.
Case 1: Last 2 digits include 0
Number options for the last 2 digits = 3. (04, 20 or 40.)
Number of options for the ten-thousands place = 4. (Any of the 4 remaining digits.)
Number of options for the thousands place = 3. (Any of the 3 remaining digits.)
Number of options for the hundreds place = 2. (Either of the 2 remaining digits.)
To combine these options, we multiply:
3*4*3*2 = 72.
Case 2: Last 2 digits do NOT include 0
Number options for the last 2 digits = 4. (12, 24, 32, or 52.)
Number of options for the ten-thousands place = 3. (Any of the remaining 4 digits but 0.)
Number of options for the thousands place = 3. (Any of the 3 remaining digits.)
Number of options for the hundreds place = 2. (Either of the 2 remaining digits.)
To combine these options, we multiply:
4*3*3*2 = 72.
Total options = 72+72 = 144.
The correct answer is E.
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To be divisible by 4, the last two digits of the number must be divisible by 4. Therefore, they can be 04, 12, 20, 24, 32, 40, and 52. We can split these into two groups: 1) 04, 20, 40, and 2) 12, 24, 32, 52VJesus12 wrote:How many five-digit numbers can be formed from the digits 0, 1, 2, 3, 4, and 5, if no digits can repeat and the number must be divisible by 4?
A. 36
B. 48
C. 72
D. 96
E. 144
The OA is the option E.
Group 1:
If the last two digits are 04, then there are 4 choices for the first (or ten-thousands) digit, 3 choices for the second (or thousands) digit, and 2 choices for the third (or hundreds) digit. So we have 4 x 3 x 2 = 24 such numbers if the last two digits are 04. Also there should be 24 numbers if the last two digits are 20 or 40. So we have 24 x 3 = 72 numbers in this group.
Group 2:
If the last two digits are 12, then there are 3 choices for the first (or ten-thousands) digit (since it can't be 0), 3 choices for the second (or thousands) digit, and 2 choices for the third (or hundreds) digit. So we have 3 x 3 x 2 = 18 such numbers if the last two digits are 12. Also there should be 18 numbers if the last two digits are 24, 32 or 52. So we have 18 x 4 = 72 numbers in this group also.
Therefore, there are a total of 72 + 72 = 144 numbers.
Answer: E
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