Every problem on the SAT can be solved without a calculator. That is what the College Board tells us, and I agree. Pretty much every single problem on the SAT can be done by your head/hand. Of course, you would like to have a calculator, and you should. Calculators make most calculations quicker, so you save time, and time is scarce on the SAT.
However, the SAT is always going to throw you a problem that cannot be done by (only) the use of a calculator.
Here's one:
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| Source |
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| I've never seen such a big number... |
OMG, what the heck is this problem? 3 to the 200th power?? That's a super large number. If you were to plug that in your calculator you'll get: 2.656139889E95.
As awesome a number that may be, chances are that you won't know what to do after that. Even if you divide that thing with 5, you'll still get a complicated huge number: 5.312279778E94.
In order to solve this problem, you'll need to know how to notice a pattern.
But before that, here's how to find the remainder in the calculator (or your hand):
1. If your quotient is a whole number, the "remainder" is 0.
2. If your quotient has a decimal, take out ONLY the decimal and multiply it with whatever number you've divided with.
3. You can use your graphing calculator's remainder function, but it won't work on this problem. I won't go over this because it's not today's lesson.
Now, for the actual tutorial:
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First, notice the first few powers of 3:
31 = 3
32 = 9
33 = 27
34 = 81
35 = 243
36 = 729
37 = 2187
38 = 6561
Pay attention to the units digit (ones place). Every four numbers follow the pattern: 3, 9, 7, 1. This sequence will continue on forever as the power of 3 increases. And because the units digit will always follow this pattern, when you divide each power of 3 by 5, your remainder will have a pattern as well. (Except for 3, because 3 is smaller than 5.)
Mind you that you can certainly solve the problem using the pattern of the remainders, however you can save time by just using the pattern of the powers of 3. All you need to know about the remainder pattern is just its existence.
In order to find the remainder of 3^200, we need to divide the exponent by the number of numbers in our pattern of 3, 9, 7, 1. Which is 4.
If we divide 200 by 4, we get 50, a whole number, meaning that the fourth/last number of our pattern shall be the unit digit of 3^200, which is 1.
If you didn't follow, then let's divide 199 by 4. What's the remainder? 3. Be careful because this 3 only means that 3^199's units digit is the third number in our pattern: 7.
198/4 's remainder is 2, meaning that 3^198 's unit digit is the second number in our pattern: 9.
197/4 's remainder is 1, meaning that 3^197 's unit digit is the first number in our pattern: 3.
Any number that is bigger than 5 and has a units digit of 1 divided by 5 is 1. Why? Just look at the following outcomes:
11 divided by 5: remainder is 1
21 divided by 5: remainder is 1
31 divided by 5: remainder is 1
Mind you that you can certainly solve the problem using the pattern of the remainders, however you can save time by just using the pattern of the powers of 3. All you need to know about the remainder pattern is just its existence.
In order to find the remainder of 3^200, we need to divide the exponent by the number of numbers in our pattern of 3, 9, 7, 1. Which is 4.
If we divide 200 by 4, we get 50, a whole number, meaning that the fourth/last number of our pattern shall be the unit digit of 3^200, which is 1.
If you didn't follow, then let's divide 199 by 4. What's the remainder? 3. Be careful because this 3 only means that 3^199's units digit is the third number in our pattern: 7.
198/4 's remainder is 2, meaning that 3^198 's unit digit is the second number in our pattern: 9.
197/4 's remainder is 1, meaning that 3^197 's unit digit is the first number in our pattern: 3.
Any number that is bigger than 5 and has a units digit of 1 divided by 5 is 1. Why? Just look at the following outcomes:
11 divided by 5: remainder is 1
21 divided by 5: remainder is 1
31 divided by 5: remainder is 1
41 divided by 5: remainder is 1
51 divided by 5: remainder is 1
and infinitely so on.
I know. I don't know what's crazier: the infinite pattern of numbers, or how we just found the remainder of 3^200 divided by 5 (a huge number) without actually dividing 3^200 at all.
Why does this work? Because math.
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| OMG it's a miracle... |
Hopefully, my little guide made sense to you. There are things in math that make sense in my head, but I don't know if I can explain it to others while making sense at the same time. After all, we are dealing with numbers, cycles, remainders, and a bunch of other arithmetic stuff, so it would be hard to explain.
If my tutorial made no sense (hopefully it did make sense), please leave a comment below as you explain with some detail as to why my explanation didn't make sense.
Oh, and if you find any errors in my post, be it mathematical, grammatical, spelling, please tell me.
Anyways, thanks for reading.
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