cyclicity

Question

find unit digit of 2^34 - 3^44.

MartyTargetTestPrep · Answer

The units digit of 2 when raised to successively higher powers goes through the cycle 2, 4, 8, 6.

So, the units digit of 2^34 can be found by dividing 34 by 4 to get 32 remainder 2 and then choosing the second units digit in the cycle, which is 4.

The units digit of 3 when raised to successively higher powers goes through the cycle 3, 9, 7, 1.

So, the units digit of 3^44 can be found by dividing 44 by 4 to get 11 remainder 0 and then choosing the fourth units digit in the cycle, which is 1.

So, the correct answer is 4 - 1 = 3

pintukr · Answer

34 = 4n + 2 (2,4,8,6)
44 = 4n (3,9,7,1)

So, 4-1 =3 (for the given expression)

EMPOWERgmatRichC · Answer

Hi darpan1234567890,

What is the source of this question?

I ask because as Ian has correctly pointed out, there is an issue with how this question is written (and it looks like we were all focused on defining the patterns behind the individual calculations along with how these types of prompts would normally be solved - and we didn't correctly note the issue with how the prompt was set up). The result of this calculation will clearly be a negative number (since 3^44 is much bigger than 2^34) and these types of prompts almost always involve a positive end result (so the negative outcome here is strange - and ultimately an unlikey situation on the Official GMAT). That having been said, I've edited the question (and corresponding answer) to account for how it might appear on the Official GMAT:

What is the units digit of 3^44 - 2^34?

The GMAT would NEVER expect you to actually calculate the value of 3^44 - 2^34, so there MUST be a pattern in these numbers that you can figure out without too much trouble. When it doubt, it's often helpful to 'list out' results until a pattern emerges.

3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81

3^5 = 243
3^6 = 729
3^7 = 2187
3^8 = 6561

Notice the pattern here? As the exponent increases, the units digit consistently follows the pattern 3, 9, 7, 1.... 3, 9, 7, 1

A similar style of pattern occurs with powers-of-2...

For example:
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16

2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256

2, 4, 8, 6.... 2, 4, 8, 6

Thus, the units digit of 3^44 is a 1 and the units digit of 2^34 is a 4.... so the units digit of that overall calculation is 1 - 4 = 7

GMAT assassins aren't born, they're made,
Rich

Contact Rich at: [email protected]

IanStewart · Answer

These solutions are not correct. The units digit of 2^34 is 4, and the units digit of 3^44 is 1. But 3^44 is clearly larger than 2^34, so 2^34 - 3^44 is a negative number. If we subtract a number ending in 1 from a number ending in 4 and end up with a negative, that negative will end in 7, not in 3 (just as 14 - 31, say, ends in 7). So the answer is 7.

I've never once seen an official GMAT question about the digits of a negative number, though, so this is almost certainly not an issue you'll encounter on the actual test.

cyclicity

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