 # Mental Math

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• Birthday 12/11/1974

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1. Burning Math
2. What is the largest number value in base-10 you can write with just 3 digits? No symbols and characters allowed. Hints: it's not 999 Ask someone to write the largest 3-digit number and they'll respond with 999. Logical answer, but we can go bigger. Some may get the "power" brainwave and think of 999 (99 to the power of 9), which calculates out as 99×99×99×99×99×99×99× 99×99. Even better is 999 (9 to the power of 99) which calculates out as 9×9×9×9×9×9×9 ... and so on 99 times. The correct answer, however, if you extend the idea even further ends up as... 99^9 (9 to the 9th power of 9). Work out the second and third powers first (9×9×9×9×9×9×9×9×9 = 387420489.) We can therefore restate the sum as 9387420489 which works out as.... very very big indeed.
3. Here is a cool trick to square numbers made up of repeating 1 (one). Count the number of repeating 1 in the number to be squared and put the digits ascending from 1 up to that number. That is, if there are four 1's in the number to be squared, put 1234. Then, put down the same numbers inversely, that is, 321. That’s it. It is so easy, it’s just mental math. Easy and fast. Let’s illustrate with an Example: If the number to be squared is 111 There are 3 repeating 1's. So, put down 123. Then, put 21 after 123. So, 1112 = 12321 See the pattern? Let's see another one, Square of 11111 =? There are 5 repeating 1's. So, put 12345. Then, put 4321 So, 111112 = 123454321
5. If you multiply two five digit numbers you can get the answer 123456789. Can you guess the two five digit numbers? This is a very beautiful puzzle and you might think that a big mathematical theory should be hidden behind it. But in fact its beauty is only incidental. Give it a try ... . . . Here is the Hint to help you solve the puzzle, 'Decompose the number (123456789) into its prime factors'. 123456789 = 3 x 3 x 3607 x 3803 Got it? OK, check it with the solution … scroll down . . . . . Find the solution url removed
6. If a number can be expressed as a product of two whole numbers, then the whole numbers are called factors of that number. In other words, A factor is a whole number which divides exactly into a whole number, leaving no remainder. For example, 13 is a factor of 52 because 13 divides exactly into 52 (52 ÷ 13 = 4 leaving no remainder). 52 = 1 x 52 = 2 x 26 = 4 x 13 So, the complete list of factors of 52 is: 1, 2, 4, 13, 26, and 52 (all these divide exactly into 52). The simple technique to find the number of factors of a given number is to express the number as a product of powers of prime numbers or prime factors. To illustrate let's find the numbers of factors of our example 52. Note that, 52 can be expressed as 4 x 13 = 22 x 13 So, the prime factors of 52 are 2 and 13. Now, increment the power of each of the prime numbers by 1 and multiply the result. In this case it will be (2 + 1) x (1 + 1) = 3 x 2 = 6 (power of 2 is 2 and power of 13 is 1) Therefore, there will 6 factors including 1 and 52. Also note that, all numbers have a factor of 1 since 1 multiplied by any number equals that number. All numbers can be divided by themselves to produce the number 1. Therefore, we normally ignore 1 and the number itself as useful factors. So, excluding, these two numbers, you will have (6 – 2) = 4 factors. To be certain the factors are: 2, 4, 13 and 26. To further illustrate let's find the numbers of factors of 48. 48 can be written as 16 x 3 = 24 x 3 So, the prime factors of 48 are 2 and 3. Now, increment the power of each of the prime numbers by 1 and multiply the result. In this case it will be (4 + 1) x (1 + 1) = 5 x 2 = 10 (the power of 2 is 4 and the power of 3 is 1) Therefore, there will be 10 factors including 1 and 48. Excluding, these two numbers, you will have (10 – 2) = 8 factors. And the factors are: 2, 3, 4, 6, 8, 12, 16 and 24.
7. Ask a friend to secretly write down ANY number (at least four digits long). e.g. 78341 Ask the friend to add up the digits... e.g. 7+8+3+4+1 = 23 ... and then subtract the answer from the first number. e.g. 78341 - 23 = 78318 Your friend then crosses out ONE digit from the answer. (It can be any digit except a zero) e.g. 7x318 Your friend then reads out what digits are left .e.g. 7-3-1-8 Even though you haven't seen any numbers, you can say what the missing digit is! EIGHT THE SECRET This great trick relies on the power of 9. How? Link removed by Moderator
8. Let’s have a game. For this trick, secretly write 73 on a piece of paper, fold it up, and give to an unsuspecting friend. Now have your friend select a four-digit number whatever (say, 3125) and enter it twice into a calculator. (31253125) Announce that the number is divisible by 137 and have him verify it on his calculator. Next, announce that he can now divide by his original four-digit number. After he has done so, dramatically command him to look at your prediction on the paper. It will match his calculator display: 73! Want to know the simple math behind this "TRICK"?
9. Here is a cool trick to add hours and minutes together: Let's try adding 3 hr 23 minutes and 2 hr 46 minutes together. Make the 3 hr 23 minutes into one number, which will give us 323 and do the same for the other number, 2 hours 46 minutes, giving us 246. Now add these two numbers together: 323 246 569 So, we have a total of 569. Now, if the last 2 digit is equal to or greater than 60, just add 40 to the total, otherwise keep it as it is. 569 + 40 = 609 And now, split the number to get the answer which is 6 hrs and 09 minutes!
10. You likely know the technique of multiplying any number by 11. Here is the trick to multiply a 2-digit and 3-digit number by 111. Multiplying a 2-digit number by 111 Example: 111 × 58 1. Write down the ones digit. > 8 2. Add the tens and ones; carry, if any. > 38 (5 + 8 = 13) 3. Repeat the previous step (and add carry, if any) > 438 (5 + 8 + 1 = 14) 4. Write the tens digit (add carry, if any) > 6438 (5 + 1 = 6) So, 111 × 58 = 6,438 Multiplying a 3-digit number by 111 Use right-to-left "sweeps" of 1, 2, 3, 2, and 1 digit. Example: 111 × 234 1. Write down the right most digit. > 4 2. Add the 2 right most digits (carry, if any) > 74 (3 + 4 = 7) 3. Add all the 3 digits (and carry, if any) > 974 (2 + 3 + 4 = 9) 4. Add the 2 left most digits (and carry, if any) > 5974 (2 + 3 = 5) 5. Write down the left most digit (add carry) > 25974 So, 111 × 234 = 25,974