Question: What is the unit’s digit of \((17^3)^4-{1973^3}^2\) ?

1. 3
2. 7
3. 15
4. 21
5. 25

Answer:

Approach Solution 1:

We know that:

(1) The unit’s digit of \((abc)^n\)is the same way as that of \(c^n\) which means that the unit’s digit of \((17^3)^4\) is the same as that of \((17^3)^4\) and the units digit of \({1973^3}^2\) is the same as that of \({3^3}^2\).

(2) If exponentiation is indicated by stacked symbols, the rule is to work from top to down, thus: \({a^m}^n=a({^m}^n)\) and not \({a(^m)}^n\) , which on the other hand equals to \(a(^{mn})\)

So:

\((7^3)^4=(7)^{3*4}=7^{12}and3^{3^2}=3(^{3^2})=3^9\)

(3) The unit’s digit of the integers in positive integer power repeats in specific pattern: The unit digit of 7 and 3 in positive integer power repeats in patterns of 4:

1. \(7^1=7\)(Last digit is 7)
2. \(7^2=49\)(Last digit is 9)
3. \(7^3=xx3\)(Last digit is 3)
4. \(7^4=xxx1\)(Last digit is 1)
5. \(7^5=xxxxxx7\)(Last digit is 7 again)

…

1. \(3^1=7\)(Last digit is 3)
2. \(3^2=9\)(Last digit is 9)
3. \(3^3=27\)(Last digit is 7)
4. \(3^4=81\)(Last digit is 1)
5. \(3^5=243\)(Last digit is 3 again)

…

Thus, the unit’s digit of \(7^{12}\) will be 1 (4th in pattern, as 12 is a multiple of cyclist number 4) and the unit’s digit of \(3^9\) will be 3 (first in pattern, as 9 = 4 * 2 + 1)

So, we have that the unit’s digit of \((17^3)^4\) = \((17)^{12}\) is 1 and the unit’s digit of \({1973^3}^2={1973^9}\) is 3. Also notice that the second number is much larger than the first one, thus their difference will be negative, something like 11 – 13 = -2, which gives the final answer that the unit’s digit of \((17^3)^4-{1973^3}^2\)is 2.

Correct option: B

“What is the unit’s digit of?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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