Question 1 - IMO B

Explanation -

Stmt 1 - @ can be anything between 0 to 4
Hence Insufficient

Stmt 2 - Clearly indicates the value of @
Hence Sufficient

Question 2 - IMO A

Explanation -

Stmt 1 -
Take some numbers such that they satisfy the given criteria.
13, 21, 29, 37, 45, 53, 61, 69, 77 ...
When any of these numbers are divided by 4, the remainder is 1

Hence Sufficient

Stmt 2 -
Take some numbers such that they satisfy the given criteria.
1^2 + 2^2 = 1 + 4 = 5 .When this is divided by 4, the remainder is 1
But 3^2 + 7^2 = 9 + 49 = 58. When this is divided by 4, the remainder is 2

Hence Insufficient

the answers are E and A respectively...

Question no.1:
3.2@#6 : If @ and # each represent single digits in the given decimal, what digit does @ represent?
(1) When the decimal is rounded to the nearest tenth, 3.2 is the result.
(2) When the decimal is rounded to the nearest hundredth, 3.24 is the result.

E---Not sufficient.
From 2 we can find # but it is not sufficient to give the result as it is less than 5.

Question no.2:
If p is a positive integer, what is the remainder when p is divided by 4?
(1) When p is divided by 8, the remainder is 5.
(2) p is the sum of the squares of two positive integers.

Statement 1 alone is sufficient as the rem will be 1 in each case.Statement 2 alone is not sufficient.Hence Ans A

I agree...My Bad....Silly Mistake
Answer for 1) is 'E'.
'@' is '1' for '#' values 5 - 9 and '@' is '2' for '#' values 0 - 4.

Solution to Question no.1:
3.2@#6 : If @ and # each represent single digits in the given decimal, what digit does @ represent?
(1) When the decimal is rounded to the nearest tenth, 3.2 is the result.
(2) When the decimal is rounded to the nearest hundredth, 3.24 is the result.

Statement 1: If the number is rounded to 3.2 then @ has to be any number between 0-4. If @ is above 4 than the number will be rounded off to 3.3 which is not the case. This statement alone is insufficient as any number between 0-4 might be the possibility.

Statement 2: If the number is rounded to 3.24 than @ can be 3 or 4. Let’s say the number is 3.23#6 if # is above 5 than the number will be rounded off to 3.24. Let’s say the number is 3.24#6 if # is below 5 than the number will be rounded off to 3.24 Hence @ can be 3 or 4. Insufficient
Even after combining both the statements it is not possible to bet on the answer.
Hence E


Solution to Question no.2:
If p is a positive integer, what is the remainder when p is divided by 4?
(1) When p is divided by 8, the remainder is 5.
(2) p is the sum of the squares of two positive integers.

Statement 1: This statement says that the number is of the form 8k+5. Now let’s divide this number by 4. 8k+5/4 = (8k/4)+(5/4)
When 8k is divided by 4 remainder is 0 and when 5 is divided by four remainder is 1. Hence the remainder when 8K+5 is divided by 4 will be 1. Statement 1 is sufficient

Statement 2: This statement alone is sufficient. You need not try different values. Since p is the sum of the square of two different numbers two different numbers can be both even, odd or one odd and one even. When both or even or odd the sum will be even hence the remainder can be 2 or 0. If one is even and one is odd the result will be odd hence the remainder will be 1 or 3. If you know this simple fact evaluating statement 2 is 10 secs job

Hence A

Let me know if you have any queries


Thanks,
Quant-Master