Sneha2021 wrote:
avigutman KarishmaBHow can I solve it logically for statement 2 without listing value of n (3,12,21,30...) and concluding that remainder could 2 (12) or 0 (30)?
Well,
Sneha2021, let's try this: if I tell you that positive integer
n is divisible by 7, can you tell me whether it's odd or even?
What if I tell you that positive integer
k has a remainder of 3 when divided by 7, a remainder of 1 when divided by 5, and a remainder of 0 when divided by 3. Now can you tell me whether
k is odd or even?
The answer to both questions above is no. We can't tell, because the terms odd and even exist only in the world of divisibility by 2, and the divisor '2' doesn't have any common factors with the divisors '3', '5', or '7'. These worlds of divisibility are completely separate from one another, so no inferences can be made about remainders going from one world to another.
By the way, my description of
k is going to fit numbers that are (3*5*7 = ) 105 apart on the number line. So, if we find possible value of
k, adding 105 over and over again will keep yielding possible values of
k. And, of course, each time we add 105, we change the value of
k back and forth between odd and even.
k could be 66, 171, 276, 381, 486, etc.
Back to your original question,
Sneha2021: yes, we can evaluate statement (2) logically, without testing cases, and without pen and paper. The information we have about the divisibility of the "certain integer" is for divisors of 9 and 2, but the question has a divisor of 5, which has no common factors with 9 or 2. Therefore, we can't infer anything about the remainder in the world of divisibility by 5.