# Practice GMAT Problem Solving Question

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What is the units digit of 6

^{15}- 7^{4}- 9^{3}?Correct Answer:

**C**- The authors of the test do not want you to do the calculations long-hand. This would consume an incredible amount of time and is absolutely unnecessary.
- In order to determine the units digit of the difference of these three terms, it is essential to know the units digit of each individual term. Since a digit raised to integer exponent follows a pattern in its units digit, you need to identify this pattern.
- In identifying the pattern of the units digit of an exponential expression, it is essential to remember that you do not need to carry out the entire multiplication process to determine the units digit of a product. Simply multiply the units digit of each number being multiplied and the units digit of this product will be the units digit of the entire expression. For example: (2389283)(24892489) will have a units digit of 7 because 9*3=27 has a units digit of 7.
- The units digit of 6 raised to an integer exponent follows a definitive pattern. Consequently, with minimal calculations, you know that the units digit of 6
^{15}is 6.

6^{1}= 6 --> units digit of 6

6^{2}= 36 --> units digit of 6

6^{3}= 216 --> units digit of 6

6^{4}= 1,296 --> units digit of 6

6^{any integer}--> units digit of 6 - The units digit of 7 raised to an integer exponent follows a definitive pattern.

7^{1}= 7 --> units digit of 7

7^{2}= 49 --> units digit of 9

7^{3}= 343 --> units digit of 3

7^{4}--> 3*7 {take units digit of 3 from 343 and multiply by 7} --> units digit of 1

7^{5}--> 1*7 {take units digit of 1 from 7^{4}and multiply by 7} --> units digit of 7

7^{6}--> 7*7 {take units digit of 7 from 7^{5}and multiply by 7} --> units digit of 9

7^{7}--> 9*7 --> units digit of 3

7^{8}--> 3*7 --> units digit of 1

Based upon this pattern, you know that the units digit of 7^{4}is 1. -
The units digit of 9 raised to an integer exponent follows a definitive pattern.

9^{1}= 9 --> units digit of 9

9^{2}= 81 --> units digit of 1

9^{3}--> 1*9 --> units digit of 9

9^{4}--> 9*9 --> units digit of 1

9^{5}--> 1*9 --> units digit of 9

9 raised to an odd integer has a units digit of 9.

Consequently, you know that the units digit of 9^{3}is 9. - Thus far you know that the units digit of 6
^{15}– 7^{4}– 9^{3}= units digit of 6 – units digit of 1 – units digit of 9. - Simplify the first two terms of this expression: units digit of 6 – units digit of 1 = units digit of 5

Note: If you are having trouble believing that this will always hold true, try a few numbers. For example, 796 - 11 = 785, 56 - 501 = -445, 86 - 271 = -185 - The expression now reads: units digit of 5 - units digit of 9.
- This is perhaps the trickiest part of the question. Some students think that since 5-9=-4, the units digit of the entire expression will be 4. However, this fails to consider that the left term could be larger than the right, resulting in a units digit of 6. For example:

15-9=6 {left term is larger}

155-99=56 {left term is larger}

155-999=-844 {right term is larger}

15-99=-84 {right term is larger} - Consequently, the crucial question in determining whether the units digit of the final expression is a 6 or a 4 is whether the left expression is larger than the right. In other words, "is (6
^{15}- 7^{4}) greater than 9^{3}?" - In order to figure this out, take an approximate guess at the value of each term. You know that 9
^{3}will be less than 1000, which is 10^{3}, so the question of whether the units digit is 6 or 4 really rests on whether 6^{15}- 7^{4}is greater than 1000 (in which case the left term will be larger than the right term and the units digit will be 6) or whether 6^{15}- 7^{4}is less than a thousand (in which case the right term will be larger than the left term and the units digit will be 4). - The test does not require long tedious calculations and these are not necessary here. It should be rather clear that 6
^{15}- 7^{4}is greater than 1000, in which case the units digit will be 6, not 4. - Units digit of 5 - units digit of 9 = units digit of 6 since the left term (i.e., the one with a 5) is larger than the right term. The final answer is a units digit of 6.
- Answer choice C is correct.
- FYI. The expression is: 470,184,984,576 – 2,401 – 729 = 470,184,981,446

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