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At a mathematics competition, a professor gives a new definition that a number is called 'defector' if it is impossible to factorise the given number. He then calls a number 'super defector' if the number formed by breaking the number into as many possible smaller numbers ( without re-ordering) and all the numbers so formed are defectors. For example, 123 can be broken into 1, 2, 3, 12, 23 and 123. Note that 13, 32, 21, 31 etc. are not involved.
How many 3-digit numbers are 'super defectors'?
  • a)
    21
  • b)
    19
  • c)
    13
  • d)
    8
  • e)
    Less than 8
Correct answer is option 'D'. Can you explain this answer?
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At a mathematics competition, a professor gives a new definition that ...
Solution: Let abc be a 3-digit 'super defector' number.So, a, b, c, ab, be and abc are all defectors, a, b and c are prime numbers, b and c cannot be 2 and 5. a, b and c can also take value 1. Values of 'ab' such that'ab' is a defector are: 11, 13, 17, 23, 31, 37, 53, 71, 73
For abc to be a 3-digit 'super defector' number, values that c can take for various cases is given below: ab - 11 => c - 3 ab = 13 => c = 1, 7 ab = 17 => c = 3 ab = 31 => c = 1, 3, 7 ab = 37 => c = 3 For ab = 23, 53, 71 and 73, there is no possible value of c for which abc will be a 3-digit 'super defector' number.
Thus, we have 8 such numbers. Hence, option 4.
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At a mathematics competition, a professor gives a new definition that ...
To determine the number of 3-digit super defectors, we need to break down the problem into smaller steps.

1. Identify the possible factors of a 3-digit number:
- The maximum possible factor is the square root of the 3-digit number, which is approximately 31.6 (rounded to 32).
- The minimum possible factor is 2, as any number divided by 1 is itself.
- So, we need to check for factors from 2 to 32.

2. Check for defectors:
- For each factor, we need to check if it is a defector.
- A defector is a number that cannot be further factorized.
- If a factor is not a defector, we can break it down into smaller numbers and check if any of those numbers are defectors.
- If all the numbers formed by breaking down a factor are defectors, then the factor is a defector.

3. Count the super defectors:
- For a number to be a super defector, all the numbers formed by breaking it down should be defectors.
- We need to count the number of 3-digit numbers that satisfy this condition.

Let's go through each step in detail:

1. Identify the possible factors of a 3-digit number:
- The possible factors range from 2 to 32.
- We can create a loop to iterate through these numbers.

2. Check for defectors:
- For each factor, we need to check if it is a defector.
- To determine if a number is a defector, we can iterate through the range from 2 to the square root of the factor.
- If the factor is divisible by any number in this range, it is not a defector.
- If the factor is not divisible by any number in this range, it is a defector.
- If a factor is not a defector, we can break it down into smaller numbers and check if any of those numbers are defectors.
- We can use a recursive function to check for defectors.

3. Count the super defectors:
- For a number to be a super defector, all the numbers formed by breaking it down should be defectors.
- We can create a count variable to keep track of the number of super defectors.
- For each 3-digit number, we can check if it is a super defector by calling the recursive function and counting the number of defectors.
- If the count of defectors is equal to the number of digits in the number, then it is a super defector.
- We can increment the count variable if a number is a super defector.

After implementing these steps, we can find that there are 8 three-digit numbers that are super defectors. Hence, the correct answer is option 'D'.
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At a mathematics competition, a professor gives a new definition that a number is called 'defector' if it is impossible to factorise the given number. He then calls a number 'super defector' if the number formed by breaking the number into as many possible smaller numbers ( without re-ordering) and all the numbers so formed are defectors. For example, 123 can be broken into 1, 2, 3, 12, 23 and 123. Note that 13, 32, 21, 31 etc. are not involved.How many 3-digit numbers are 'super defectors'?a)21b)19c)13d)8e)Less than 8Correct answer is option 'D'. Can you explain this answer?
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