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A pattern is a repetition of numbers that results from repeating calculations. Sometimes you can find patterns in questions that otherwise seem impossible to solve.
What is the units digit of 3801?
| Solution
You know you don't have time to calculate. This is a reasoning question. There must be a shortcut.
Try to find a pattern starting with smaller exponents. The pattern shows up pretty quickly.
31 = 3 |
35 = 3 × 81 = 243 |
32 = 9 |
36 = 3 × 243 = 729 |
33 = 27 |
37 = 3 × 729 = 2187 |
34 = 3 × 27 = 81 |
38 = 3 × 2187 = 6561 |
The units digit repeats after "4 powers." The pattern is clearest for exponents that are multiples of 4.
Every exponent that is a multiple of 4 has a units digit of 1. Since 800 is a multiple of 4, the units digit of 3800 is 1. Looking at the pattern, the units digit of 3801 will be 3.
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800score Tip:
When you see a question where you scratch your head and think it will take an hour to solve... chances are that there is a shortcut. Using Backsolve, Plug-In, Find Patterns or math shortcuts, you will likely find the easy way to solve it.
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What is the sum of integers from 1 to 100, inclusive?
| Solution
Look for a pattern starting with smaller groups of numbers.
1 + 2 + 3 + 4 = 1 + 2 + 3 + 4 = 5 + 5 = 10
1 + 2 + 3 + 4 + 5 + 6 = 1 + 2 + 3 + 4 + 5 + 6 = 7 + 7 + 7 = 21
First trick: Add first and last.
Second trick: How many pairs are there for an even number of integers?
You can check the pattern by doing one more batch of numbers.
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = (1 + 10)(10/2) = (11)(5) = 55
So the sum of the integers from 1 to 100 is (1 + 100) (100/2) = (50)(101) = 5,050 |
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