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MrWhite
If integer p is a positive factor of 36 and integer t is a positive factor of 27, which of the following products CANNOT be equal to \(p^2 - t^2\) ?

(A) (12)(6)
(B) (19)(17)
(C) (39)(13)
(D) (45)(27)
(E) (63)(9)


Peculiarity about 27 is that all the positive factors except 1 are multiples of 3 as \(27=3^3\).
36 may or may not have factors that are multiple of 3.
Straightforward C, because both p-t and p+t will either be multiples of 3 or not be multiplies of 3.
In C, one of them is a multiple of 3 and other not.


Reason is as given below.

Scenario 1: Both p and t are factors of 3
Here, both (p-t) and (p+t) will be multiples of 3.
Example A, D and E
Scenario 2: p is not a factor of 3 and t is a factor of 3
Here, both (p-t) and (p+t) will not be multiples of 3.
Scenario 3: t is 1
So (p-1)(p+1), that is a product of two consecutive odd or consecutive even integers.
Example: B
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Short cut => p-t + p + t = 2p
so sum of all options should be a max 72 and a factor of it. If not the answer choice is incorrect. This should make you go faster.
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MrWhite
If integer p is a positive factor of 36 and integer t is a positive factor of 27, which of the following products CANNOT be equal to \(p^2 - t^2\) ?

(A) (12)(6)
(B) (19)(17)
(C) (39)(13)
(D) (45)(27)
(E) (63)(9)

Responding to a pm: Yes, it is a tough question. Though they haven't put in the convoluted option, but you cannot depend on that while solving it.

The first thing to do here is to realize that any option that gives the product as a multiple of 2 but not 4 will not work. Why?
Here is a blog post that explains why: https://anaprep.com/number-properties-a ... m-product/
But we do not get the answer here since all options are either odd or a product of 4.

Next I would find the factors of 36 and 27 and parse through the options to see if they can be made. I would not assume that the given options are of the form (p+t)(p-t) because it is not justified.
For example, if an option gave us 27*21, does it mean that p and t are 24 and 3? 24 is not a factor of 36 so does it mean that p^2 - t^2 cannot be equal to 27*21? No. Because 27*21 = 63*9 which is a valid option. So that approach is not appropriate.

36 = 1, 2, 3, 4, 9, 12, 18, 36
27 = 1, 3, 9, 27

(A) (12)(6)
We have p = 9 and t = 3 possible so move on.

(B) (19)(17)
We have p = 18 and t = 1 possible so move on.

(C) (39)(13)
39*13 = 3*13*13 so only possible values of (p+t)(p-t) is 39*13. 169*3 is not possible because 86 and 83 cannot be the factors because they are too large.
But 39 and 13 cannot be the factors either because 26 and 13 are not factors of 36 and 27.
Hence this is the answer

(D) (45)(27)
We have p = 36 and t = 9 possible so move on.

(E) (63)(9)
We have p = 36 and t = 27 possible so move on.

Answer (C)
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MrWhite
If integer p is a positive factor of 36 and integer t is a positive factor of 27, which of the following products CANNOT be equal to \(p^2 - t^2\) ?

(A) (12)(6)
(B) (19)(17)
(C) (39)(13)
(D) (45)(27)
(E) (63)(9)

Approach 1
p^2-t^2 = (p+t)(p-t)

that means each of the given answer choices can be represented in this form
(A) (12)(6)
p+t = 12
p-t = 6
Solving p = 9 t = 3
satisfies both (p factor of 36 t factor of 27) Eliminate
(B) (19)(17)

Solving like above p=18, t =1. Eliminate
(C) (39)(13)
p = 26, Does not satisfy, p is not a factor of 36 correct answer
(D) (45)(27)
(E) (63)(9)

Approach 2
Factors of 36 1,2,3,4,6,9,12,18,36
Factors of 27 1,3,9,27

(A) (12)(6)
(9+3)(9-3) Eliminate
(B) (19)(17)
(18+1) (18-1) Eliminate
(C) (39)(13)
Can't factorize this like above in terms of factors keep
(D) (45)(27)
(36+9)(36-9) Eliminate
(E) (63)(9)
(36+27)(36-27) Eliminate
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The key identity here is p2 - t2 = (p + t)(p - t). Each answer choice is already written as a product of two numbers, so we can set those equal to (p + t) and (p - t), then solve for p and t.

First, note that:
- Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
- Factors of 27: 1, 3, 9, 27

Now check each choice:

A: (12)(6) → p + t = 12, p - t = 6 → p = 9, t = 3. Is 9 a factor of 36? Yes. Is 3 a factor of 27? Yes. WORKS.

B: (19)(17) → p + t = 19, p - t = 17 → p = 18, t = 1. Is 18 a factor of 36? Yes. Is 1 a factor of 27? Yes. WORKS.

C: (39)(13) → p + t = 39, p - t = 13 → p = 26, t = 13. Is 26 a factor of 36? NO. Is 13 a factor of 27? NO. To be thorough, we check all other factor pairs of 507 (= 3 × 132): (169, 3) gives p = 86, t = 83; (507, 1) gives p = 254, t = 253. None of these are factors of 36 or 27. DOES NOT WORK.

D: (45)(27) → p + t = 45, p - t = 27 → p = 36, t = 9. Is 36 a factor of 36? Yes. Is 9 a factor of 27? Yes. WORKS.

E: (63)(9) → p + t = 63, p - t = 9 → p = 36, t = 27. Is 36 a factor of 36? Yes. Is 27 a factor of 27? Yes. WORKS.

Answer: C

General principle: Whenever you see a2 - b2, immediately think (a + b)(a - b). This converts a subtraction problem into a system of equations that's much easier to test.
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