If a and b are positive integers is a < 20 < b < 30?

From statement 1)

ab = 22 this would mean a could be 1 and b is 22 or vice versa or a is 2 and b is 11 etc.. since we get different answers it is insufficient.

Statement 2)

a < 9 < b

a satisfies the condition that it is less than 20, but b could be anything including 31 so it is insufficient.

Now if we combine both.

a could be 1,2,11,22 from statement 1)
b could be 1,2,11,22 from statement 1)

Now if a < 9 then it could be 1 or 2

if it is 1 then b is 22 as a result this provides an answer of yes

a = 1 < 20 < b = 22 < 30 , yes

However, if a is 2 then b is 11

a = 2 < 20 < b = 11 < 30 , no.

Since we get two answers then the answer choice is E.

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#1
22; 22*1 or 2*11
so relation a < 20 < b < 30 is both yes and no
insufficient
#2
a^2 < 81 < b^2
again a=1 ,b =11
or a=2, b=10
insufficient
IMO E

A fairly simple question on Numbers, this is a question which tests your knowledge of properties of numbers.

Be careful not to assume that a<20<b<30; you are trying to find out if a<20<b<30.

Also remember that if a and b are positive integers, \(a^2\) and \(b^2\) will be perfect squares. Perfect squares are always squares of integers.
With this in mind, let us proceed to evaluate the statements.

From statement I alone and the question data, we can say that if ab = 22, then the possible values of a and b are:
a = 2 and b = 11 and
a = 1 and b = 22.

In the first case, a<20 but b>20. So we get a NO as an answer. In the second case, a<20<b<30, so we get a YES as an answer. Clearly, statement I is insufficient.
Answer options A and D can be ruled out; possible answer options are B, C and E.

From statement II alone and the question data, we know that a<9 and b>9. Although a<20, we cannot say whether b will always be between 20 and 30. Statement II is insufficient. Answer option B can be eliminated.

When we combine the two statements, we have,
a = 2 and b = 11 &
a = 1 and b = 22.
In both the cases, \(a^2\)<81<\(b^2\). We saw earlier that, when we used these values, we got a YES and NO situation. Clearly, that won’t change now and the combination of statements is still insufficient.
The correct answer option is E.

Hope this helps!

Spoiler also indicates OA is E.

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