Among all the students who tried to answer the questions presented last week, Anirudh Matange (Class VI, Ahlcons School) gave the best explanations. The answers are:

Q1. The Commutative Law for multiplication.

Q2. The Prime Numbers $(2, 3, 5, 7, 11, 13, 17, 19)$ give rise to only two rectangles:

Q3. The largest number of rectangles arise from $12, 18,$ and $20.$

Q4. The rectangles with side $2$ can be formed by the Even Numbers: $2, 4, 6, \dots, 20.$

Q5. All the Multiples of $3$ (namely: $3, 6, \dots, 18$) can form rectangles with side $3.$

Here is another mathematical activity on the concept of divisibility. When a number $n$ divides a number $m$ evenly, we say that $m$ is divisible by $n$, or that $n$ is a factor of $m$. There are a number of tests that determine whether a given number is divisible by $2, 3, 5,$ or $9.$ By doing the following experiments, you can discover these tests for yourself.

1. Consider the numbers: $2030, 4201, 89782, 129083, 124, 5435, 67656, 9087, 8888, 90919.$ For each number, you have to tell the last digit when the given number is multiplied by $2$.

2. For each of the numbers in Activity 1, you have to tell the last digit when the given number is multiplied by $5$.

3. Make a table with 2 columns. In the first column, place the multiples of $9$ less than $100$ ($9, 18, 27, 36,\dots , 81, 90$). In the second column, note the sum of their digits.

4. Make a table with two columns. In the first column, put any $10$ multiples of $3$ less than $100$—such as $12, 15, 63, 51$—that are not all multiples of $9$. In the second column, note the sum of their digits.

Now use your experiments and guess the answer of the following:

Q1. Which of the following numbers is divisible by $2$: $100000, 1201, 2342, 9083, 2124, 21245, 1906, 6757, 1978, 9879.$

Q2. Which of the following numbers is divisible by $5$: $100000, 1201, 2342, 9083, 2124, 21245, 1906, 6757, 1978, 9879.$

Q3. What is the rule for checking whether a number is divisible by $9$?

Q4. Which of the following numbers is divisible by $3$: $101010, 1201, 20112, 2124, 21223, 1906, 6757, 1978, 9879, 1080.$

Bright students should verify, and then prove that their guesses are correct. But the proofs, while easy, are not covered in school syllabi. For the moment, it is enough to know the tests of divisibility without knowing why they work.