Problems involving prime and composite numbers are often not too hard. It’s good to know prime numbers at least till 50, or even better, 100.
As a recap, the prime numbers till 50 are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
Q. Amy is given a number x. She picks two numbers a and b and then generates a sequence of numbers x, x +a, x + a + b, x + 2a + b, x + 2a + 2b, i.e. she alternately adds ato get a new number and then bto get another number. Which two numbers should she choose for aand b so that she is guaranteed to generate a prime number, regardless of the choice of x?
a.) 2, 3
b) 3, 6
c) 2, 4
d) 7, 14
So we don’t want to get stuck with composite numbers alone. This happens if aand bhave a common factor. For example if a = 3 and b = 6, then picking x = 3 leaves us with composite numbers alone. So option a is the correct answer.
Q. There are k students in a class. What could be the value of kif every attempt to sort them perfectly into teams of greater than 2, but less than k fails?
k is clearly a prime. So the answer is 17.
This was a really easy question. A slightly hard question would be something like this:
Q. Which of the following could be a value of k if khas an odd number of factors?
You can factorise each number above. But a useful trick is to observe that factors come in pairs. So if a is a factor of n, so is n/a. So factors will be counted in pairs, unless we have a repeated factor, i.e. a factor a such that a=n/a. Or we need a2=n i.e. n should be a perfect square. So the answer is b.
This is a useful fact. In the previous question, the answer choices were small. But imagine if the same question were rephrased as follows.
Q. Which of the following could be a value of kif khas an odd number of factors?
The answer is a because 100000000 is a perfect square.
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