Bunuel wrote:
How many five-digit positive integers can be formed using each of the digits from {1, 2, 3, 5} at least once such that they are a multiple of 15?
A. 36
B. 24
C. 18
D. 15
E. 12
Are You Up For the Challenge: 700 Level QuestionsNote: The question says that you should use 1, 2, 3, 5 at least once; it does not say that you should use only those digits
To make a number a multiple of 15, the number must be a multiple of 5, which, in this case, only possible if the units digit is 5 or 0
Also, the number must be a multiple of 3 => Sum of the digits is a multiple of 3
Since we need to use all the digits at least once, we already have a 4-digit number with sum of digits 1+2+3+5 = 11
If we wish to use the digit 0 at the end (units digit), the digits would be 1, 2, 3, 5, 0 => sum of the digits is 11, not divisible by 3. Hence, 0 cannot be used.
Thus, the possibilities are (keeping 5 fixed at the end):
# 1, 2, 3, 1, 5 => sum = 12 => Number of ways = 4!/2! = 12
# 1, 2, 3, 4, 5 => sum = 15 => Number of ways = 4! = 24
# 1, 2, 3, 7, 5 => sum = 18 => Number of ways = 4! = 24
Thus, we can form 12 + 24 + 24 = 60 such numbers
Now: This does not match any of the answer options.
So I am thinking that the question actually wanted to say that you can use ONLY the digits 1,2,3,5 at least once.
In that case, the answer is 12 (option E)
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Sujoy Kumar Datta |
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