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GMAT Date: 02-18-2019
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WE:Engineering (Consulting)
Re: For any positive number n, the function #n represents the value of the
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12 Jun 2019, 01:40
Official explanation:
Steps 1 & 2: Understand Question and Draw Inferences
Given:
Let k = n + a.bcde. . . , where n is a multiple of 10 and a, b, c, d, e etc. denote the units, tenths, hundredths, thousandths, ten-thousandths digits of k respectively.
Example: Say k is equal to 123.789. We can also write this value as 120 + 3.789. This form is comparable to n + a.bcde . . .
So, n denotes the value of all the digits at the tens, hundreds, thousands and higher place values (if k = 123.789, n = 120; if k = 3456.987, n = 3450; if k = 43789.34, n = 43780 and so on)
We have split k into n and a.bcde. . . because the question doesn’t indicate whether the magnitude of k is in tens or hundreds or thousands etc. (that is, is k a number like 12.345 or like 123.789 or like 3456.987 etc.) This knowledge is not important either because all the action is happening at the units digit and the digits to the right of the decimal point. So, we’ve simply expressed the value of all the digits at the tens, hundreds, thousands and higher place values in a single term n
#k =
n + a, if b < 5
n + (a+1) if b ≥ 5
To find:
The units digit of #k
Since n is a multiple of 10, from the expression of #k, we can say that the units digit of #k =
a, if b < 5
a + 1, if b ≥ 5
Thus, to answer the question, we need to know the value of a and whether b < 5 or not
Step 3: Analyze Statement 1 independently
(1) #(10k) = 10k
10k = 10n + ab.cde. . .
Since a is now the tens digit and b is the units digit, we can write
10k = 10n + 10a + b.cde. . .
10k = 10(n+a) + b.cde. . .
Applying the definition of the function #n, we can write:
#(10k) =
10(n+a) + b if c < 5
10(n+a) + (b+1) if c ≥ 5
We are given that #(10k) = 10k.
#(10k) = 10(n+a) + b.cde . . .
If c < 5, then by substituting the value of #(10k) in the above equation, we get:
10(n+a) + b = 10(n+a) + b.cde. . .
b = b + 0.cde. . .
This implies, c = d = e . . .= 0
Thus, k is a number of the form n + a.b, where b is the sole digit after the decimal point
If c ≥ 5, then by substituting the value of #(10k) in the above equation, we get:
10(n+a) + (b+1) = 10(n+a) + b.cde. . .
b +1 = b + 0.cde . . .
1 = 0.cde . . .
For no values of digits c, d, e . . . will the above equation be satisfied.
Therefore, this case is not possible
Thus, from Statement 1, we conclude that k is a number of the form n + a.b, where b is the sole digit after the decimal point
However, we still don’t know the value of a or whether b is less than 5 or not.
So, Statement 1 alone is not sufficient.
Step 4: Analyze Statement 2 independently
(2) #(100k) is 10300.
100k = 100n + abc.de. . .
100k = (100n + 100a + 10b) + c.de. . .
Applying the definition of the function #n, we can write:
#(100k) =
(100n + 100a + 10b) + c if d < 5
(100n + 100a + 10b) + (c+1) if d ≥ 5
We are given that #(100k) = 10300
If d < 5, then we can write
(100n + 100a + 10b) + c = 10300
Remember that in the above expression, a, b and c are digits (therefore, lie between 0 and 9, inclusive) whereas n is a multiple of 10.
Comparing the units digits on both sides of the equation, we get: c = 0
Comparing the tens digits on both sides of the equation, we get: b = 0
Comparing the hundreds digits on both sides of the equation, we get: a = 3
Since b < 5 in this case, the units digit of #k = a = 3
If d ≥ 5, then we can write
(100n + 100a + 10b) + (c+1) = 10300
Comparing the units digits on both sides of the equation, we get: c + 1 = 0
So, c = 9 and 1 is carried over to tens place
Comparing the tens digits on both sides of the equation, we get: b + 1 = 0 (Note, we’re writing b+1 and not b due to the carry-over)
So, b = 9 and 1 is carried over to hundreds place
Comparing the hundreds digits on both sides of the equation, we get: a + 1 = 3
So, a = 2
Since b > 5 in this case, the units digit of #k = a +1 = 3
Thus, we see that both possible values of #(100k) lead to the same value of the units digit of #k
Thus, Statement 2 is sufficient to find a unique answer to the question.
Step 5: Analyze Both Statements Together (if needed)
Since we’ve already arrived at a unique answer in Step 4, this step is not required
Answer: Option B
its very lengthy.. looking for simple explanation