The question is from CAT Number Theory - Remainders. It discusses about co-prime numbers and remainders. A range of CAT questions can be asked based on concepts from Remainders. When we add two numbers that are not coprime, the sum of these two numbers cannot be prime. Awesome intuitive stuff, but very often forgotten. Also, learn something from Euler today.
Question 7: A prime number p greater than 100 leaves a remainder q on division by 28. How many values can q take?
q can be 1.
If q =2, number would be of the form 28n + 2 which is a multiple of 2.
Similarly, when q = 4, number would be of the form 28n + 4 which is again a multiple of 2. Any number of the form 28n + an even number will be a multiple of 2.
When q = 7, number would be of the form 28n + 7 which is a multiple of 7.
So, the only remainders possible are remainders that share no factors with 28. Or numbers that are co-prime to 28.
There is a formula for this and a shorter way of finding the number of numbers co-prime to a given natural number. A more detailed discussion on this is provided here and here.
1, 3, 5, 9, 11, 13, 15, 17, 19, 23, 25 and 27. q can take 12 different values.
The question is "How many values can q take?"
Choice B is the correct answer.
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