In life, we often see certain patterns repeat over and over again. After all, if everything in life were unpredictable, we’d have a hard time forecasting tomorrow’s weather or how long it will take to go to work next week. Luckily, many patterns repeat in recurring, predictable patterns. A simple example is a calendar. If tomorrow is Friday, then the following day will be Saturday, and Sunday comes afterwards (credit: Rebecca Black). Moreover, if today is Friday, then 7 days from now will also be Friday, and 70 days from now will also be Friday, and onwards ad infinitum (even with leap years). These patterns are what allow us to predict things with 100% certainty.
Some patterns are inexact, or can change dramatically based on external factors. If you think of the stock market or the weather, people often have a general sense of prediction but it is hardly an exact science. Some patterns are more rigid, but can still fluctuate a little. Your work schedule or the weekly TV guide tend to remain the same for long stretches of time, but are not always exactly the same year over year. Finally, there are patterns that never change, like the Earth’s rotation or the number of days in a year (accounting for the dreaded leap year). These patterns are rigid, and can be forecasted decades ahead of time.
On the CAT, this same concept of rigid prediction is utilized to solve mathematical questions that would otherwise require a calculator. A common example would be to ask for the unit digit of a huge number, as something like 15^16 is far too large to calculate quickly on exam day, but the unit digit pattern can help provide the correct answer. Given any number that ends with a 5, if we multiply it by another number that ends with a 5, the unit digit will always remain a 5. This pattern will never break and will continue uninterrupted until you tire of calculating the same numbers over and over. A similar pattern exists for all numbers that end in 0, 1, 5 or 6, as they all maintain the same unit digit as they are squared over and over again.
For the other six digits, they all oscillate in predetermined patters that can be easily observed. Taking 2 as an example, 2^2 is 4, and 2^3 is 8. Afterwards, 2^4 is 16, and then 2^5 is 32. This last step brings us back to the original unit digit of 2. Multiplying it again by 2 will yield a unit digit of 4, which is 64 in this case. Multiplying by 2 again will give you something ending in 8, 128 in this case. This means that the units digit pattern follows a rigid structure of 2, 4, 8, 6, and then repeats again. So while it may not be trivial to calculate a huge multiple of 2, say 2^150, its unit digit can easily be calculated using this pattern.
Let’s look at a problem that highlights this pattern recognition nicely:
What is the units digit of (13)^4 * (17)^2 * (29)^3?
There is no reasoning, no shrewdness, required to solve this with a calculator. You punch in the numbers, hope you don’t make a typo and blindly return whatever the calculator displays without much thought. However, if you’re forced to think about it, you start extrapolating the patterns of the unit digit and the general number properties you can use to your advantage.
For starters, you are multiplying 3 odd numbers together, which means that the product must be odd. Given this, the answer cannot possibly be answer choice C, as this is an even number. We’ve managed to eliminate one answer choice without any calculations whatsoever, but we may have to dig a little deeper to eliminate the other three.
Firstly, recognize that the unit digit is interesting because it truncates all digits other than the last one. This means this is the same answer as a question that asks: (3^4) * (7^2) * (9^3). While we could conceivably calculate these values, we only really need to keep in mind the unit digit. This will help avoid some tedious calculations and reveal the correct answer much more quickly.
Dissecting these terms one by one, we get:
3^4, which is 3*3*3*3, or 9*9, or 81.
7^2, which is just 49.
9^3, which is 9*9*9, or 81 * 9, or 729.
The fact that we truncated the first digit of the original numbers changes nothing to the result, but does serve to make the calculations slightly faster. Furthermore, we can truncate the tens and hundreds digits from this final calculation and easily abbreviate:
81 * 49 * 729 as
1 * 9 * 9.
This result again gives 81, which has a units digit of 1. This means that the correct answer ends up being answer choice E. It’s hard to see this without doing some calculations, but the amount of work required to solve this question correctly is significantly less than what you might expect at first blush. An unprepared student may approach it by calculating 13^4 longhand, and waste a lot of time getting to an answer of 28,561. (What? You don’t know 13^4 by heart?) Especially considering that the question only really cares about the final digit of the response, this approach is clearly more dreary and tedious than necessary.
The units digit is a favorite question type on the CAT because it can easily be solved by sound reasoning and shrewdness.