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| (|a| + 4)*(|b| - 3) | = 16. How many pairs of integers (a,

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| (|a| + 4)*(|b| - 3) | = 16. How many pairs of integers (a,  [#permalink]

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New post 05 Aug 2019, 10:18
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A
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C
D
E

Difficulty:

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Question Stats:

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| (|a| + 4) * (|b| - 3) | = 16. How many pairs of integers (a,b) can satisfy this equation ?

A. 10
B. 14
C. 16
D. 18
E. 20

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| (|a| + 4)*(|b| - 3) | = 16. How many pairs of integers (a,  [#permalink]

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New post Updated on: 21 Aug 2019, 12:21
2
(|a|+4) ≥4 but (|b|-3) can be negative or positive in any combination in order to satisfy | (|a| + 4) * (|b| - 3) | = 16.

(|a|+4) can take three values: 4, 8 and 16. Correspondingly, (|b|-3) must pair with following values: 4, +/-2, +/-1. Note that (|b|-3) = -4 is not possible because |b|≥0.

|(|a| + 4) * (|b| - 3)|
=|4*4|... a=0,b=+/-7, number of solution=1*2=2
=|8*2|...a=+/-4,b=+/-5, number of solution=2*2=4
=|8*-2|...a=+/-4,b=+/-1, number of solution=2*2=4
=|16*1|...a=+/-12,b=+/-4, number of solution=2*2=4
=|16*-1|...a=+/-12,b=+/-2, number of solution=2*2=4

Thus, total number of pairs of integers (a,b) that satisfy the equation = 2+4+4+4+4=18
Answer is (D)

Originally posted by chondro48 on 05 Aug 2019, 10:40.
Last edited by chondro48 on 21 Aug 2019, 12:21, edited 1 time in total.
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Re: | (|a| + 4)*(|b| - 3) | = 16. How many pairs of integers (a,  [#permalink]

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New post 07 Aug 2019, 02:58
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1
In a question on Absolute values, remember that you need to think about the input values that the can be put inside the ‘Mod’ and not the output. The output is anyways going to be positive, since the absolute value function is a distance function.

In this question also, note that you can plug in both positive and negative values for a and b, without affecting the dynamics of the equation.
The question also gives us that elbow space to try only integer values, otherwise, this could have been a more difficult question.

The product of the expression inside the modulus should give us a +16 or a -16. So, essentially, we have to look for the factors of 16.

16 can be written as a product of 2 integral factors in the following ways:
16 = 16 * 1
16 = 8 * 2
16 = 4 * 4
16 = 2 * 8
16 = 1 * 16.

Let’s take the first case. As per this, |a| + 4 = 16 and |b| - 3 = 1. For the above equations, there will be two values of a and b, which will satisfy the individual equations i.e. a = 12 or -12 and b = 4 or – 4.

So, this case gives us 4 pairs.
Similarly, cases 2, 4 and 5 will give us 4 pairs.

For case 4, |a| + 4 = 4 and |b| - 3 = 4. Only one value of a i.e. a=0 satisfies the first equation, while, the second equation will be satisfied by two values of b i.e. b = 7 or -7.
So, case 4 gives us 2 pairs i.e. (0, 7) and (0,-7).

Therefore, the total number of integral pairs of (a,b) is 18.
The correct answer option is D.

Hope this helps!
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Re: | (|a| + 4)*(|b| - 3) | = 16. How many pairs of integers (a,   [#permalink] 07 Aug 2019, 02:58

| (|a| + 4)*(|b| - 3) | = 16. How many pairs of integers (a,

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