Guna wants to put 12 figs in each box and Anshu wants to put 7 figs in each box. How many arrangements are possible? Think and find out the different ways how — 1. Guna can arrange 12 figs in a rectangular manner. 2. Anshu can arrange 7 figs in a rectangular manner.
Answer
A rectangular arrangement of rows × columns is possible only when rows × columns gives the number of figs. So we simply need the pairs of factors.
1. Guna's 12 figs — 12 = 1 × 12 = 2 × 6 = 3 × 4, and each can be turned around:
Six rectangular arrangements of 12 figs. If you count a rectangle and its turned-around twin as the same, there are three different shapes.
7 figs can be laid out in only one way — a single line, standing up or lying down.
7 × 1 or 1 × 7 → only one shape
Why the difference: 12 has six factors (1, 2, 3, 4, 6, 12) so it gives six arrangements, but 7 has only two factors (1 and 7) so it gives just one line. This single fact is what makes 7 a prime and 12 a composite number.
Q2.
Observe the number of rows and columns in each of the arrangements. How are they related to 12?
Answer
In every arrangement, rows × columns = 12. So the number of rows and the number of columns are always factors of 12.
Rows
12
6
4
3
2
1
Columns
1
2
3
4
6
12
Rows × columns
12
12
12
12
12
12
Reading the top row we get 12, 6, 4, 3, 2, 1 — exactly the list of factors of 12.
Tip: this gives a lovely way to find all factors of a number — count how many different rectangles you can build with that many counters or tamarind seeds.
Q3.
How many prime numbers are there from 21 to 30? How many composite numbers are there from 21 to 30?
Answer
Test each of the ten numbers from 21 to 30.
Number
21
22
23
24
25
26
27
28
29
30
A factor other than 1 and itself
3
2
—
2
5
2
3
2
—
2
Prime / composite
C
C
P
C
C
C
C
C
P
C
Primes from 21 to 30 → 23 and 29 — that is 2 primes
Composites from 21 to 30 → 21, 22, 24, 25, 26, 27, 28, 30 → 8 composite numbers
2 + 8 = 10, and there are exactly 10 numbers from 21 to 30. ✔
Q4.
Can we list all the prime numbers from 1 to 100? Here is an interesting way to find prime numbers. Just follow the steps given and see what happens.
Answer
Yes. Write 1 to 100 in a grid and follow the five steps — cross out 1, then circle 2 and cross out its later multiples, circle 3 and cross out its later multiples, then 5, then 7. After that everything left is already circled.
The Sieve of Eratosthenes. Only the circled numbers survive — those are the primes.
Tip: you can stop crossing out after 7. Any composite number below 100 must have a factor of 10 or less (because 11 × 11 = 121 is already more than 100), so it is caught by 2, 3, 5 or 7.
Q5.
Guna and Anshu started wondering how this simple method is able to find prime numbers! Think how this method works. Read the steps given above again and see what happens after each step is carried out.
Answer
The sieve works because of one simple idea: a composite number always has a smaller prime as a factor, so it is bound to get crossed out at some step.
Follow what happens to a number, say 91:
91 is odd → survives step 2 (multiples of 2)
9 + 1 = 10, not a multiple of 3 → survives step 3
does not end in 0 or 5 → survives step 4
but 91 = 7 × 13 → crossed out when the multiples of 7 go
And a prime such as 37:
37 is not a multiple of 2, 3 or 5, and 6 × 6 = 36 < 37 < 49 = 7 × 7
so no number from 2 to 6 divides it → 37 is never crossed out → it stays circled
The reason in one line: each step removes the numbers that have that particular prime as a factor. Since every composite number has at least one prime factor, every composite gets removed. The numbers that stay behind have no prime factor smaller than themselves — and that is exactly what being prime means.
Did you know? Eratosthenes lived about 2200 years ago in Greece. He also measured the size of the Earth using shadows — and his answer was surprisingly close to the correct value.