## Thursday, June 13, 2002

### Experience Mathematics # 2 - Divisibility

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.

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 $3$: $101010, 1201, 20112, 2124, 21223, 1906, 6757, 1978, 9879, 1080.$