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16 Sep 2014, 01:24
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What is the units digit of \((17^3)^4-1973^{3^2}\)?
A. 0
B. 2
C. 4
D. 6
E. 8
A. 0
B. 2
C. 4
D. 6
E. 8
16 Sep 2014, 01:24
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Official Solution:
What is the units digit of \((17^3)^4-1973^{3^2}\)?
A. 0
B. 2
C. 4
D. 6
E. 8
MUST KNOW FOR THE GMAT:
I. The units digit of \((abc)^n\) is the same as that of \(c^n\).
Hence, the units digit of \((17^3)^4\) is the same as that of \((7^3)^4\) and the units digit of \(1973^{3^2}\) is the same as that of \(3^{3^2}\).
II. \({(a^m)}^n=a^{mn}\) and \(a^{m^n}=a^{(m^n)}\).
Hence, \({(7^3)}^4=7^{(3*4)}=7^{12}\) and \(3^{3^2}=3^{(3^2)}=3^9\).
III. The units digit of integers in positive integer powers repeats in a specific pattern called cyclicity. The units digit of 7 and 3 in positive integer powers 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. \(3^1=3\) (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!)
...
Hence, the units digit of \(17^{12}\) is 1, as the 12th number in the repeating pattern 7-9-3-1 is 1, and the units digit of \(1973^9\) is 3, as the 9th number in the repeating pattern 3-9-7-1 is 3.
BACK TO THE QUESTION:
Thus, we know that the units digit of \({(17^3)}^4=17^{12}\) is 1, and the units digit of \(1973^{3^2}=1973^9\) is 3. If the first number is greater than the second number, the units digit of their difference will be 8, for example, 11 - 3 = 8. However, if the second number is greater than the first number, the units digit of their difference will be 2, for example, 11 - 13 = -2.
Consider this, even if the first number were \(100^{12}\) instead of \(17^{12}\), and the second number were \(1000^9\) instead of \(1973^9\), the first number, \(100^{12}=10^{24}\), would still be smaller than the second number \(1,000^9=10^{27}\). Therefore, \(17^{12} < 1973^9\), and the units digit of the difference is 2.
Answer: B
What is the units digit of \((17^3)^4-1973^{3^2}\)?
A. 0
B. 2
C. 4
D. 6
E. 8
MUST KNOW FOR THE GMAT:
I. The units digit of \((abc)^n\) is the same as that of \(c^n\).
Hence, the units digit of \((17^3)^4\) is the same as that of \((7^3)^4\) and the units digit of \(1973^{3^2}\) is the same as that of \(3^{3^2}\).
II. \({(a^m)}^n=a^{mn}\) and \(a^{m^n}=a^{(m^n)}\).
Hence, \({(7^3)}^4=7^{(3*4)}=7^{12}\) and \(3^{3^2}=3^{(3^2)}=3^9\).
III. The units digit of integers in positive integer powers repeats in a specific pattern called cyclicity. The units digit of 7 and 3 in positive integer powers 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. \(3^1=3\) (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!)
...
Hence, the units digit of \(17^{12}\) is 1, as the 12th number in the repeating pattern 7-9-3-1 is 1, and the units digit of \(1973^9\) is 3, as the 9th number in the repeating pattern 3-9-7-1 is 3.
BACK TO THE QUESTION:
Thus, we know that the units digit of \({(17^3)}^4=17^{12}\) is 1, and the units digit of \(1973^{3^2}=1973^9\) is 3. If the first number is greater than the second number, the units digit of their difference will be 8, for example, 11 - 3 = 8. However, if the second number is greater than the first number, the units digit of their difference will be 2, for example, 11 - 13 = -2.
Consider this, even if the first number were \(100^{12}\) instead of \(17^{12}\), and the second number were \(1000^9\) instead of \(1973^9\), the first number, \(100^{12}=10^{24}\), would still be smaller than the second number \(1,000^9=10^{27}\). Therefore, \(17^{12} < 1973^9\), and the units digit of the difference is 2.
Answer: B
shasadou
Joined: 12 Aug 2015
Last visit: 24 Nov 2022
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Concentration: General Management, Operations
Schools: Johnson '18 (WL) Duke '18 (M) Tuck '18 (D)
GMAT 1: 640 Q40 V37
GMAT 2: 650 Q43 V36
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WE:Management Consulting (Consulting)
04 Feb 2016, 05:05
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when a question trhows at u a trick like the one above with the second number larger than the first one try to deconstruct the ambiguity with simple numbers. If 111-1973 then the results ends in 2. If 11111 - 1973 ---> 8.
