CAT Practice : Number System

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Product of two consecutive integers is always even. When will this be a multiple of 3 as well.

Multiples

    Q.5: n2 + 5n + 6 is a multiple of 6. n is natural number less than 100. How many values can n take?
    1. 33
    2. 65
    3. 66
    4. 67

 

  • Correct Answer
    Choice C. 66

Explanatory Answer

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Detailed Solution

n2 + 5n + 6
(n + 2) (n + 3)
(n + 2) and (n + 3) are two consecutive numbers.
One of (n + 2) or (n + 3) has to be even.

We need n such that (n + 2) (n + 3) is a multiple of 3.
(n + 2) or (n + 3) should be a multiple of 3.
n leaves a remainder of 0 or 1 when divided by 3.
n could be 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18 ....
So, from among the first 99 natural numbers, N cannot take 2, 5, 8, 11, 14…….98.

There are 33 numbers in this list – we are effectively listing all numbers {3 × 0 + 2, 3 × 1 + 2, 3 × 3 + 2, ….3 × 32 + 2}.

So, N can take 99 – 33 = 66 values.
Answer choice (c)

Correct Answer: 66 values



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More questions from Number System

  1. Product of digits
  2. Sum of first n natural numbers
  3. Perfect Square
  4. Last two digits
  5. Multiples
  6. Greatest integer function
  7. Medians, Number
  8. 44th digit of a number
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