Consider any 3-digit number, say (). Make two other 3-digit numbers from these digits by cycling these digits around, yielding and . Now add the three numbers. Using algebra, justify that the sum is always divisible by 37. Will it also always be divisible by 3? [Hint: Look at some multiples of 37.]

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Question 11

Consider any 3-digit number, say $abc$ ( $100a + 10b + c$ ). Make two other 3-digit numbers from these digits by cycling these digits around, yielding $bca$ and $cab$ . Now add the three numbers. Using algebra, justify that the sum is always divisible by 37. Will it also always be divisible by 3? [Hint: Look at some multiples of 37.]

Solution

We will use algebra to represent the 3-digit numbers and then add them together.

Step 1 — Expressing the Numbers and Finding Their Sum

Let us take any 3-digit number. We can write its digits as $a$ , $b$ , and $c$ . Here, $a$ is the hundreds digit, $b$ is the tens digit, and $c$ is the units digit. The value of this number is $100a + 10b + c$ .

The first number is $abc$ : $100a + 10b + c$

Now, we cycle the digits to make two more numbers. The second number is $bca$ : $100b + 10c + a$

The third number is $cab$ : $100c + 10a + b$

Let us add these three numbers together. $\text{Sum} = (100a + 10b + c) + (100b + 10c + a) + (100c + 10a + b)$

Now, we group the terms with $a$ , $b$ , and $c$ together. $\text{Sum} = (100a + a + 10a) + (10b + 100b + b) + (c + 10c + 100c)$

Let us add the like terms. $\text{Sum} = 111a + 111b + 111c$

We can see that 111 is a common factor in all terms. $\text{Sum} = 111(a + b + c)$

$\boxed{\text{Sum} = 111(a + b + c)}$

Step 2 — Checking for Divisibility

From Step 1, we found that the sum of the three numbers is $111(a + b + c)$ . To check if the sum is divisible by 37, we need to see if 111 is divisible by 37. Let us divide 111 by 37. $111 \div 37 = 3$ This means that $111 = 3 \times 37$ .

So, we can write the sum as: $\text{Sum} = (3 \times 37)(a + b + c)$ This shows that 37 is a factor of the sum. Therefore, the sum is always divisible by 37.

Now, let us check for divisibility by 3. Since $111 = 3 \times 37$ , we already know that 3 is a factor of 111. So, the sum $111(a + b + c)$ can also be written as $3 \times 37 \times (a + b + c)$ . This clearly shows that 3 is a factor of the sum. Therefore, the sum is always divisible by 3.

Answer

(i) The sum is always divisible by 37. (ii) The sum is also always divisible by 3.

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What can you say about the numbers in the pyramid and the number at the top in the following cases?

(i) The first four Virahāṅka-Fibonacci numbers are written in the bottom row of a four row pyramid. (ii) The first 29 Virahāṅka-Fibonacci numbers are written in the bottom row of a 29 row pyramid.

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Q10

In the trick given above, instead of finding the difference of the two 2-digit numbers, find their sum. What will happen? For example:

We start with 31. After reversing we get 13. Adding 31 and 13, we get 44.
We start with 28. After reversing we get 82. Adding 28 and 82, we get 110.
We start with 12. After reversing we get 21. Adding 12 and 21, we get 33.

Observe that all these numbers are divisible by 11. Is this always true? Can we justify this claim using algebra?

Q11

Q12

Consider any 3-digit number, say $abc$ . Make it a 6-digit number by repeating the digits, that is $abcabc$ . Divide this number by 7, then by 11, and finally by 13. What do you get? Try this with other numbers. Figure out why it works. [Hint: Multiply 7, 11 and 13.]

Q13

There are 3 shrines, each with a magical pond in the front. If anyone dips flowers into these magical ponds, the number of flowers doubles. A person has some flowers. He dips them all in the first pond and then places some flowers in shrine 1. Next, he dips the remaining flowers in the second pond and places some flowers in shrine 2. Finally, he dips the remaining flowers in the third pond and then places them all in shrine 3. If he placed an equal number of flowers in each shrine, how many flowers did he start with? How many flowers did he place in each shrine?

Q14

A farm has some horses and hens. The total number of heads of these animals is 55 and the total number of legs is 150. How many horses and how many hens are on the farm?

Can you solve this without letter-numbers?

[Hint: If all the 55 animals were hens, then how many legs would there be? Using the difference between this number and 150, can you find the number of horses?]

Q15

A mother is 5 times her daughter's age. In 6 years' time, the mother will be 3 times her daughter's age. How old is the daughter now?

Q16

Two friends, Gauri and Naina, are cowherds. One day, they pass each other on the road with their cows. Gauri says to Naina, "You have twice as many cows as I do". Naina says, "That's true, but if I gave you three of my cows, we would each have the same number of cows". How many cows do Gauri and Naina have?

Q17

I run a small dosa cart and my expenses are as follows:

Rent for the dosa cart is ₹5000 per day.
The cost of making one dosa (including all the ingredients and fuel) is ₹10.

(i) If I can sell 100 dosas a day, what should be the selling price of my dosa to make a profit of ₹2000?

(ii) If my customers are willing to pay only ₹50 for a dosa, how many dosas should I aim to sell in a day to make a profit of ₹2000?

Q18

Evaluate the following sequence of fractions:

$\frac{1}{3}, \frac{(1 + 3)}{(5 + 7)}, \frac{(1 + 3 + 5)}{(7 + 9 + 11)}$

What do you observe? Can you explain why this happens?

[Hint: Recall what you know about the sum of the first $n$ odd numbers.]

Q19

Karim and the Genie

Karim was taking a nap under a tree. He had a dream about a magical lamp and a genie. He heard a voice saying, “I have come to serve you, Oh master”. He woke up and to his surprise, it was a genie!

“Do you want to make money?”, asked the genie. Karim nodded dumbly in bewilderment. The genie continued, “Do you see the banyan tree over there? All you have to do is go around it once. The money in your pocket will double”.

Karim immediately started towards the tree, only to be stopped by the genie. “One moment!”, said the genie. “Since I am bringing you great riches, you should share some of your gains with me. You must give me 8 coins each time you go around the tree.”

Thinking that was a trifling amount, Karim readily agreed.

He went around the tree once. Just as the genie had said, the number of coins in his pocket doubled! He gave 8 coins to the genie. He made another round. Again the number of coins doubled. He gave 8 more coins to the genie. He went around the tree for the third time. The number of coins doubled again, but to his horror, he was left with only 8 coins, exactly the number of coins he owed the genie!

As Karim began to wonder how the genie tricked him, the genie let out a loud laugh and disappeared.

(i) How many coins did Karim initially have?

(ii) For what cost per round should Karim agree to the deal, if he wants to increase the number of coins he has?

(iii) Through its magical powers, the genie knows the number of coins that Karim has. How should the genie set the cost per round so that it gets all of Karim's coins?