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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