Std 6 Mathematics · Chapter 4

4. Patterns

Solutions to Practice Set 4.1, Practice Set 4.2 and Exercise 4. Maharashtra State Board (Balbharati), Grade Six Mathematics.

About the chapter

Numbers often follow a rule. Such a rule makes a pattern. In this chapter, we learn about prime, composite, coprime and twin prime numbers. We also study patterns like triangular numbers, square numbers, cube numbers and Virahanka numbers.

Key points

A prime number has only two divisors, 1 and the number itself.
A composite number has more than two divisors. 1 is neither prime nor composite.
2 is the smallest prime number. It is the only even prime number.
Two numbers whose only common divisor is 1 are coprime numbers.
To find the next number in a series, first find the rule of the pattern.

Formulas and terms

Twin primes – two prime numbers whose difference is 2, like 11 and 13.
Triangular numbers – 1, 3, 6, 10, 15, …
Square numbers – 1, 4, 9, 16, 25, … (1 × 1, 2 × 2, 3 × 3, …)
Cube numbers – 1, 8, 27, 64, … (1 × 1 × 1, 2 × 2 × 2, …)
Virahanka numbers – 0, 1, 1, 2, 3, 5, 8, … Each number is the sum of the two numbers before it.

Practice Set 4.1

1Fill in the blanks.

(i) 1 is neither prime nor composite.
(ii) 2 is the smallest prime number.
(iii) 2 is the only even prime number.
(iv) 3 is the smallest odd prime number.

2Write five pairs of coprime numbers between 1 and 50.

Coprime numbers have only 1 as their common divisor.

Five pairs of coprime numbers are: (4, 9), (8, 15), (14, 25), (21, 22), (35, 48).

Other answers are also possible, like (9, 16) or (13, 40).

3Complete the pattern of prime numbers: 2, __, 5, 7, __, 13, 17, __, 23, 29, __, 37, __, __, 47.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

4Identify the 'twin prime' numbers from the following and write the reason.

Twin primes are prime numbers whose difference is 2.

(i) 3 and 5 – Twin primes. Both are prime and 5 − 3 = 2.
(ii) 7 and 9 – Not twin primes. 9 is not a prime number.
(iii) 11 and 13 – Twin primes. Both are prime and 13 − 11 = 2.
(iv) 23 and 29 – Not twin primes. Their difference is 6.
(v) 29 and 31 – Twin primes. Both are prime and 31 − 29 = 2.
(vi) 41 and 47 – Not twin primes. Their difference is 6.

Practice Set 4.2

1Identify the pattern in each number series and write the next three numbers.

(1) 1, 1, 1, 1, 1, 1, … – The number 1 repeats. Next: 1, 1, 1
(2) 1, 2, 3, 4, 5, 6, 7, … – Add 1 each time. Next: 8, 9, 10
(3) 1, 3, 5, 7, 9, … – Odd numbers. Add 2 each time. Next: 11, 13, 15
(4) 2, 4, 6, 8, 10, 12, … – Even numbers. Add 2 each time. Next: 14, 16, 18
(5) 1, 3, 6, 10, 15, 21, … – Triangular numbers. Add 2, 3, 4, 5, … Next: 28, 36, 45
(6) 1, 4, 9, 16, 25, 36, … – Square numbers. Next: 49, 64, 81
(7) 1, 8, 27, 64, 125, … – Cube numbers. Next: 216, 343, 512
(8) 0, 1, 1, 2, 3, 5, 8, 13, 21, … – Virahanka numbers. Add the two numbers before. Next: 34, 55, 89
(9) 1, 2, 4, 8, 16, 32, 64, … – Multiply by 2 each time. Next: 128, 256, 512
(10) 1, 3, 9, 27, 81, … – Multiply by 3 each time. Next: 243, 729, 2187

Exercise 4

1Complete the table below (the dot figures with numbers 2, 5, 10, 17, …).

Each figure is a square of dots with one extra dot. So each number is one more than a square number.

1 × 1 + 1

2 × 2 + 1

3 × 3 + 1

4 × 4 + 1

5 × 5 + 1

6 × 6 + 1

Figure	1st	2nd	3rd	4th	5th	6th
Square number	1 × 1 = 1	2 × 2 = 4	3 × 3 = 9	4 × 4 = 16	5 × 5 = 25	6 × 6 = 36
Number in table	2	5	10	17	26	37

Rule: Number = square number + 1

2The figures show hexagonal numbers. What is the next number in this figure? Show it with the help of the figure.

The given figures have 1, 7, 19 and 37 dots. Each new figure adds one ring of dots.

1 + 6 = 7
7 + 12 = 19
19 + 18 = 37
37 + 24 = 61

The next hexagonal number: 61 dots arranged as a hexagon (37 + 24 = 61).

The next hexagonal number is 61.

3Observe the number pattern and write the next three numbers: 2, 9, 28, 65, …

Each number is one more than a cube number.

1 × 1 × 1 + 1 = 2
2 × 2 × 2 + 1 = 9
3 × 3 × 3 + 1 = 28
4 × 4 × 4 + 1 = 65
5 × 5 × 5 + 1 = 126
6 × 6 × 6 + 1 = 217
7 × 7 × 7 + 1 = 344

The next three numbers are 126, 217 and 344.

4Write the first five triangular numbers. What is the tenth triangular number?

The first five triangular numbers are 1, 3, 6, 10, 15.

10th triangular number = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55

The tenth triangular number is 55.

5What number series is formed if the counting numbers are added in ascending order?

1 = 1
1 + 2 = 3
1 + 2 + 3 = 6
1 + 2 + 3 + 4 = 10
1 + 2 + 3 + 4 + 5 = 15

The series 1, 3, 6, 10, 15, … is formed. It is the series of triangular numbers.

6Write the sequence of numbers formed by adding the odd numbers starting from 1.

1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25

The sequence 1, 4, 9, 16, 25, … is formed. It is the series of square numbers.

7What is the sum of 1 + 2 + 3 + … + 50 + 49 + … + 3 + 2 + 1?

Look at the pattern:

1 + 2 + 1 = 4 = 2 × 2
1 + 2 + 3 + 2 + 1 = 9 = 3 × 3
1 + 2 + 3 + 4 + 3 + 2 + 1 = 16 = 4 × 4

So, 1 + 2 + 3 + … + 50 + 49 + … + 2 + 1 = 50 × 50 = 2500

The sum is 2500.

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