Answer to Puzzle 5

COFFIN

Amazing number 73

73 x 14 = 1022. From the product find the sum of square of last two digits and sqaure of remaining digits.
10^2 + 22^2 = 584 = 73 x 8. Repeat the above for the product 584
5^2 + 84^2 = 7081 = 73 x 97. Repeat the above for the product 7081
70^2 + 81^2 = 11461 = 73 x 157 . Repeat the above for product
114^2 + 61^2 = 16717 = 73 x 229 and so on...
167^2 + 17^2 = 28178 = 73 x 386
281^2 + 78^2 = 85045 = 73 x 1165
850^2 + 45^2 = 724525 = 73 x 9925 and so on......
So all sums will have 73 as one of the factor. Amazing is it not! But then why this happens?
Can we prove it? (Another one of 73 is 73 x 137 = 10001; 100^2 + 01^2 = 10001 = 73 x 137)

Puzzle 5

The maker does not want it.
The buyer does not use it;
The user does not see it;
What is it?

Answer to Puzzle 4

Take a number. Say ABCD. It can be written as 1000A+100B+10C+D
= 999A + A + 99B + B + 9C +C + D
= 9(111A+11B+C) + A+B+C+D
In the above the first term 9 (111A+11B+C) is divisible by 9 obviously since 9 is a factor.
So if the remaining terms A+B+C+D are divisible by 9 then the entire
number is divisible by 9. So the divisibility rule for nine is if the sum of the digits of a given
number is divisible by 9 then the entire number is divisible by 9. the same rule applies for
3 also and the proof is same as above. For 6 also the same proof is applicable and the rule is the number should be an even number and the sum of the digits shall be divisible by 3.

Puzzle 4: Prove Divisibility Rule of 9

How to find out whether any given number is divisible by 9?

Most of you know. Just add all the digits in the number and

if the sum is divisible by 9 then entire number is divisible by 9.

The Question is how do you prove that mathematically?

Scratch your brain or wait for the solution in my next posting.

Answer to Puzzle 3

1. The third. Lions that haven't eaten in three years are dead.
2. The woman was a photographer. She shot a picture of her husband, developed it, and hung it up to dry.
3. Freeze them first. Take them out of the jugs, and put the ice in the barrel. You will be able to tell which water came from which jug.
4. The answer is Charcoal.
5. Sure you can: Yesterday, Today, and Tomorrow!

Well Done Sriram! You have answered four out of five except Q.3. Good Show

Puzzle 3 (containing 5 puzzles)

1. A murderer is condemned to death. He has to choose between 3 rooms.The first is full of raging fires, the second is full of assassins withloaded guns, and the third is full of lions that haven't eaten in threeyears. Which room is safest for him?
2. A woman shoots her husband. Then she holds him under water for overfive minutes. Finally, she hangs him. But, five minutes later, they bothgo out together and enjoy a wonderful dinner together. How can this be?
3. There are two plastic jugs filled with water. How could you put allof this water into a barrel, without using the jugs or any dividers, andstill tell which water came from which jug?
4. What is black when you buy it, red when you use it, and grey whenyou throw it away?
5. Can you name three consecutive days without using the words Monday,Tuesday, Wednesday, Thursday, Friday, Saturday, or Sunday?
Answer in the next posting. You may give your answer in the comment section in the meantime if you have solved.

Answer to Puzzle/Problem 2

7 * 11 * 13 = 1001. When you multiply 1001 with any 3 digit number it repeats itself.

That is why all 3 digit numbers when repeated itself to make 6 digit numbers, they

are all divisible by 7, 11 and 13.

Problem No. 2

Write a 3 digit number. Repeat the same number by the side so that it becomes a 6 digit number. i.e. Supposing you wrote a number 407 then you write the same 407 again so that the resultant 6 digit number is 407407. Now you may use a calculator if you are not good at dividing.

The 6 digit number you have written is cleanly divisible by 7 without any remainder. (for example 407407 is divisble by 7 and leaves a quotient of 58201). Check it out for the your number and find the quotient. Now the resultant Quotient is divisible by 11 without any remainder. (58201 is divisble by 11 and the quotient is 5291). Check it out for your number and find the quotient again. Now the resultant quotient is divisible by 13 without any remainder. (5291 is divisible by 13 and the quotient is 407!). Check it out and are you surprised to find out the resultant quotient is the 3 digit number which originally you wrote? Nice, is n't it? Now the question is Why does this happen for any 3 digit number ? Prove it.

Answer to Puzzle 1

Find out the lowest number that will be divisible by numbers 1 t0 10, using LCM (L0west common Multiple) which you studied in school. Any number that is divisible by 10,9,8,7 will we be divisible automatically by 2,3,4,5 and 6. So LCM of 10,9,8, and 7 is 2*5*9*4*7 = 2520 which is the lowest number that will be divisible by all the numbers from 2 to 10.
So one number less i.e. 2519 will always leave a remainder which will be one number less than divider. Very simple is n't it!

Puzzle/Problem No. 1

Let me start with a simple mathematics question.
What is the significance of the number "2519"? The answer is:
This is the smallest number which when divided by 10 leaves a remainder of 9
when divided by 9 gives a remainder of 8, when divided by 8 gives a remainder of 7,
When divided by 7 leaves 6 as remainder, when divided by 6 leaves 5 as remainder,
when divided by 5 leaves 4 as remainder, when divided by 4 leaves 3 as remainder,
when divided by 3 leaves 2 as remainder and when when divided by 2 leaves 1 as remainder.
Interesting! is n't it? Now the question is how do you arrive at the number?
Pl. try your self and if not successful wait for my next blog where I will give answer.