If x is an integer from 41 to 80, what is the probabi

Question

If  x  is an integer from 41 to 80, what is the probability that  x^3 - 2x^2 - 3x  is a multiple of 40?

A) 9/40
B) 1/4
C) 3/10
D) 3/8
E) 2/5

kevincan · Answer

The first step is probably to factor the expression

paragw · Answer

Here's my solution

Riyaagoell · Answer

Requesting a way to solve this question. How to approach such questions?

HarshavardhanR · Answer

https://gmatclub.com/forum/download/file.php?id=89812 ---- 40 = 8 x 5 . So, we need at least 3 "2"s and at least one "5" from (x-3)(x)(x+1). Case 1: x is even Observe -> if x is even, then the other 2 terms are odd. Then, the only way to get 3 "2"s is if x is a multiple of 8 . But remember, this is not the only condition. We also have to get "5". So, why not quickly check all the 8 multiples for cases where the product will contain a 5 as well? (x-3) x (x+1) Counted? 45 48 Yes - contains both 5 and 8 53 56 57 No 64 65 Yes - contains both 5 and 8 69 72 73 No 80 Yes - contains both 5 and 8 # Cases to be counted = 3 (48, 64, 80) Case 2: x is odd In this case, the other two terms are even. But if the other two terms are even numbers that are not multiple of 4 (like 46 or 42), then, again, the product of the three terms cannot contain 8. Observe the example in the diagram for reference. So, - (x-3) and (x+1) are 4-multiples (of the form 4k) - Therefore, x is of the form 4k+3 Again, remember, this alone is not sufficient. We also need to ensure that the product contains a 5. (x-3) x (x+1) Counted? 40 43 44 Yes 44 47 48 No 48 51 52 No 52 55 56 Yes 56 59 60 Yes 60 63 64 Yes 64 67 68 No 68 71 72 No 72 75 76 Yes 76 79 80 Yes # Cases to be counted = 6 (43, 55, 59, 63, 75, 79) Total # Favorable cases = 6 + 3 = 9 Total # outcomes = 40 (# numbers from 41 to 80) The required Probability = 9/40. Choice A . --- Harsha GMAT-Club-Forum-r0hkr34i.png

egmat · Answer

Hi Riyaagoell,

Good question. The reason this one looks scary is the range ( 41  to  80 ) and the messy cubic, but there's a clean  4 -step routine that cracks it. paragw's solution follows exactly this path - let me lay out the  thinking  behind it so you can repeat it.

Step 1 - Factor the expression

Never test a raw cubic. Pull out the common x:

x3 − 2x2 − 3x = x(x2 − 2x − 3) = x(x − 3)(x + 1)

Now you're asking: when is  x(x − 3)(x + 1)  divisible by  40 ?

Step 2 - Break 40 into coprime pieces

40 = 8 × 5 , and  8  and  5  share no factors. So the product must be divisible by  5  and  by  8  - handle each separately, then combine.

Divisible by 5:  look at the three factors mod  5 . The product is a multiple of  5  exactly when one factor is. That happens at x ≡ 0, 3, 4 (mod 5) -  3  winning remainders.

Divisible by 8:  just test x =  0  through  7  in the product. Only x ≡ 0, 3, 7 (mod 8) work -  3  winning remainders.

Step 3 - Combine

Since  5  and  8  are coprime, every pairing of a mod- 5  winner with a mod- 8  winner gives one distinct remainder mod  40 . So the favorable remainders are  3 × 3 = 9  out of the  40  possible remainders mod  40 .

Step 4 - Use the range cleverly

Here's the elegant part:  41  to  80  is  40  consecutive integers, which is exactly one complete cycle of remainders mod  40 . So each remainder from  0  to  39  shows up  exactly once . That means  9  favorable remainders =  9  favorable values.

Probability =  9/40  -  therefore  answer A.

The big takeaways:  factor first ,  split the modulus into coprime factors , and remember that any block of  n  consecutive integers hits each remainder mod  n  exactly once - that last trick is what saves you from testing all  40  numbers.

Answer: A

Riyaagoell · Answer

Thank you for explaining it so well. I am still unable to understand step 4. If possible can you please help me understand the range?

If x is an integer from 41 to 80, what is the probabi

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(x-3)	x	(x+1)	Counted?
45	48		Yes - contains both 5 and 8
53	56	57	No
	64	65	Yes - contains both 5 and 8
69	72	73	No
	80		Yes - contains both 5 and 8

GMAT Problem Solving (PS) Questions

If x is an integer from 41 to 80, what is the probabi

Prep Toolkit

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