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State Board · Class 6 · Mathematics

Lesson 4: Patterns

5 concepts · 40 practice questions with answers · Free to read

This lesson is also in: मराठी

Prime, composite, twin and coprime numbers

From daily life

Seven marbles can be arranged in only one row, but eight marbles can be made into 2 rows of 4. Some numbers cannot be broken into equal rows, and these are special.

Simple definition

A prime number (मूळ संख्या) has exactly two divisors, 1 and itself. A composite number (संयुक्त संख्या) has more than two divisors.

Rule 1: Prime and composite numbers — Primes up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. Composite numbers: 4, 6, 8, 9, 10, … The number 1 is neither prime nor composite. The number 2 is the smallest prime and the only even prime. The number 3 is the smallest odd prime.

123456789101112131415161718192021222324252627282930

In the figure : In the figure the gold squares in the grid are the prime numbers up to 30: 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. The grey square is 1, which is neither prime nor composite.

  • Easy : Easy: 13 is prime, because only 1 and 13 divide it.
  • Medium : Medium: 21 is composite, because 21 = 3 × 7.
  • Exam-type : Exam: 2 is the only even prime, because every other even number is also divisible by 2.

Rule 2: Twin primes and coprime numbers — Two primes whose difference is 2 are twin primes (जुळ्या मूळ संख्या): 5 and 7, 17 and 19, 41 and 43. There are 8 pairs of twin primes between 1 and 100. Two numbers with only 1 as a common divisor are coprime (सहमूळ): 9 and 14.

  • Easy : Easy: 11 and 13 are twin primes (13 − 11 = 2).
  • Medium : Medium: 23 and 29 are not twin primes, since the difference is 6.
  • Exam-type : Exam: 21 and 22 are coprime, because consecutive numbers have only 1 as a common divisor. But 12 and 18 are not coprime (GCD is 6).
Live figure

Prime, composite, twin and coprime numbers

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Remember

1: neither. 2: only even prime. Twin primes differ by 2. Coprime: GCD 1.

Common mistake

Calling 1 a prime number. It has only one divisor.

Shortcut

Memory clue: "मूळ = दोनच भाजक; सहमूळ = मसावि एक".

Spot the difference

TypeExample
Prime2, 3, 5, 7, 11
Composite4, 6, 8, 9, 10
Twin primes17 and 19
Coprime9 and 14

Practice questions

Easy
  1. 1Is 17 a prime number?
  2. 2Is 27 a prime number?
  3. 3Which is the smallest prime number?
Medium
  1. 4Is 1 prime or composite?
  2. 5Find the twin prime partner of 41.
  3. 6Are 6 and 35 coprime?
Challenging
  1. 7Are 12 and 21 coprime?
  2. 8Why is 2 the only even prime number?
Show answers (try them yourself first)
  1. 1Yes.
  2. 2No, 27 = 3 × 9.
  3. 32.
  4. 4Neither.
  5. 543.
  6. 6Yes, GCD is 1.
  7. 7No, GCD is 3.
  8. 8Every other even number has 2 as a divisor in addition to 1 and itself.

Even, odd, Virahanka and triangular numbers

From daily life

Bowling pins are set in rows of 1, 2, 3 and 4, making a triangle of 10 pins. Number patterns like this are found everywhere.

Simple definition

Triangular numbers (त्रिकोणी संख्या): 1, 3, 6, 10, 15, … Virahanka numbers (विरहांक संख्या): 0, 1, 1, 2, 3, 5, 8, 13, … where each is the sum of the two before it.

Rule 1: Triangular numbers — They come from 1, 1 + 2, 1 + 2 + 3, 1 + 2 + 3 + 4, … = 1, 3, 6, 10, … The 7th triangular number is 1 + 2 + … + 7 = 28. The sum of two consecutive triangular numbers is a square: 10 + 15 = 25 and 15 + 21 = 36.

1st2nd3rd4th

In the figure : In the figure the dot patterns show the triangular numbers 1, 3, 6 and 10. In every pattern the gold row is the new row of dots that is added.

  • Easy : Easy: The next triangular number after 15 is 15 + 6 = 21.
  • Medium : Medium: 21 + 7 = 28 is the next one.
  • Exam-type : Exam: The 8th triangular number is 1 + 2 + … + 8 = 36, and 36 is also a square (6 × 6).

Rule 2: Virahanka numbers — 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, … Each is the sum of the two before it: 5 + 8 = 13, 8 + 13 = 21. Squares of sides 1, 1, 2, 3, 5 fit together one after another, and the next square has side 3 + 5 = 8.

11235

In the figure : In the figure squares with sides 1, 1, 2, 3 and 5 (the Virahanka numbers) fit together one after the other. The next square would have the side 3 + 5 = 8.

  • Easy : Easy: After 13 and 21 comes 13 + 21 = 34.
  • Medium : Medium: After 21 and 34 comes 55.
  • Exam-type : Exam: Which of 21, 22, 34 are Virahanka numbers? 21 and 34.
Live figure

Even, odd, Virahanka and triangular numbers

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Remember

Triangular: add 1, 2, 3, …. Virahanka: add the last two.

Common mistake

Starting Virahanka numbers with 1, 2. They start 0, 1, 1, 2.

Shortcut

Memory clue: "त्रिकोणी = वाढती बेरीज; विरहांक = मागच्या दोनांची बेरीज".

Spot the difference

SeriesFirst terms
Triangular1, 3, 6, 10, 15, 21
Virahanka0, 1, 1, 2, 3, 5, 8

Practice questions

Easy
  1. 1Find the triangular number after 15.
  2. 2What is the 5th triangular number?
  3. 3What comes after 8 and 13 in the Virahanka numbers?
Medium
  1. 4What comes after 21 and 34?
  2. 5Is 15 + 21 a square number?
  3. 6What is the 8th triangular number?
Challenging
  1. 7Show that 10 + 15 is a square number.
  2. 8Is 20 a triangular number?
Show answers (try them yourself first)
  1. 121.
  2. 21 + 2 + 3 + 4 + 5 = 15.
  3. 321.
  4. 455.
  5. 5Yes, 36 = 6 × 6.
  6. 636.
  7. 725 = 5 × 5.
  8. 8No, between 15 and 21 there is none.

Square and cube numbers

From daily life

Tiles can be laid in a square, like 4 rows of 4 tiles. A cube of sugar has the same length, width and height. Such numbers have names: square and cube numbers.

Simple definition

A square number (वर्गसंख्या) is a number multiplied by itself: 1 × 1, 2 × 2, 3 × 3, … A cube number (घनसंख्या) is a number multiplied by itself three times: 1 × 1 × 1, 2 × 2 × 2, …

Rule 1: Square numbers — 1, 4, 9, 16, 25, 36, 49, … Adding the first odd numbers gives squares: 1 + 3 + 5 + 7 = 16 and the first eight odd numbers add up to 64. Also 1 + 2 + 1 = 4, 1 + 2 + 3 + 2 + 1 = 9, and 1 + 2 + … + 10 + … + 2 + 1 = 100.

1st2nd3rd4th

In the figure : In the figure the dot patterns show the square numbers 1, 4, 9 and 16. Each time a gold L-shaped layer with the next odd number of dots is added.

  • Easy : Easy: 4 × 4 = 16.
  • Medium : Medium: 1 + 3 + 5 + 7 + 9 = 25, which is 5 × 5.
  • Exam-type : Exam: 1 + 2 + … + 12 + … + 2 + 1 = 12 × 12 = 144.

Rule 2: Cube numbers — 1, 8, 27, 64, 125, 216, … A cube with edge 3 is made of 3 × 3 × 3 = 27 small cubes. The number 64 is both a square (8 × 8) and a cube (4 × 4 × 4).

ABC

In the figure : In the figure the cubes A, B and C have 1, 2 and 3 small cubes along an edge, so they are made of 1, 8 and 27 small cubes. These are cube numbers.

  • Easy : Easy: 3 × 3 × 3 = 27.
  • Medium : Medium: 6 × 6 × 6 = 216.
  • Exam-type : Exam: A cube of edge 5 contains 5 × 5 × 5 = 125 small cubes.
Remember

Square = n × n. Cube = n × n × n. Sum of first n odd numbers = n².

Common mistake

Thinking 3 squared is 6. It is 3 × 3 = 9.

Shortcut

Memory clue: "वर्ग = दोनदा, घन = तीनदा" (square = twice, cube = three times).

Spot the difference

TypeFirst numbers
Square1, 4, 9, 16, 25, 36
Cube1, 8, 27, 64, 125

Practice questions

Easy
  1. 1Find 7 × 7.
  2. 2Find the square of 12.
  3. 3Find the sum of the first five odd numbers.
Medium
  1. 4Find the sum of the first nine odd numbers.
  2. 5Find 1 + 2 + 3 + 4 + 3 + 2 + 1.
  3. 6Find the cube of 5.
Challenging
  1. 7How many small cubes make a cube of edge 6?
  2. 8Name a number below 100 that is both a square and a cube other than 1.
Show answers (try them yourself first)
  1. 149.
  2. 2144.
  3. 31 + 3 + 5 + 7 + 9 = 25.
  4. 49 × 9 = 81.
  5. 54 × 4 = 16.
  6. 6125.
  7. 76 × 6 × 6 = 216.
  8. 864 = 8² = 4³.

Number series

From daily life

If you see 5, 10, 15, 20, you can say the next number is 25 without counting. Finding the rule of a series lets us predict what comes next.

Simple definition

A number series (संख्या मालिका) follows a rule. To find the next numbers, find the rule: add a fixed number, multiply by a fixed number, or a special pattern.

Rule 1: Common series — 5, 10, 15, 20 → 25, 30, 35. Triangular 1, 3, 6, 10, 15, 21, 28 → 36, 45, 55. Squares 1, 4, 9, 16, 25, 36, 49 → 64, 81, 100. Cubes 1, 8, 27, 64, 125 → 216, 343, 512. Doubling 3, 6, 12, 24 → 48, 96, 192.

  • Easy : Easy: 10, 20, 30, 40 → 50, 60, 70.
  • Medium : Medium: 1, 4, 9, 16, 25 → 36, 49, 64.
  • Exam-type : Exam: Tripling 2, 6, 18, 54 → 162, 486, 1458.

Rule 2: Hidden rules — Virahanka 0, 1, 1, 2, 3, 5 → 8, 13, 21. Cubes minus 1: 0, 7, 26, 63 → 124, 215, 342, because 5³ − 1 = 124, 6³ − 1 = 215 and 7³ − 1 = 342.

3×26×212×2?×248

In the figure : In the figure each number is the one before it multiplied by 2, so the missing number is 12 × 2 = 24.

  • Easy : Easy: 3, 6, 9, 12 → 15, 18, 21.
  • Medium : Medium: 1, 3, 5, 7 → 9, 11, 13.
  • Exam-type : Exam: 0, 7, 26, 63: each is a cube minus 1, so the next numbers are 124, 215, 342.
Remember

Find the rule first: add, multiply, square, cube, or add the last two.

Common mistake

Checking only the first two numbers to guess the rule. Test the rule on all numbers.

Shortcut

Memory clue: "नियम शोधा, मग पुढे जा".

Spot the difference

SeriesRule
5, 10, 15+ 5
3, 6, 12, 24× 2
2, 6, 18, 54× 3

Practice questions

Easy
  1. 1Next three: 3, 6, 9, 12, …
  2. 2Next three: 1, 3, 5, 7, …
  3. 3Next three: 20, 22, 24, …
Medium
  1. 4Next three: 2, 4, 8, 16, …
  2. 5Next three: 1, 3, 9, 27, …
  3. 6Next three: 1, 3, 6, 10, 15, 21, …
Challenging
  1. 7Next three: 1, 4, 9, 16, 25, 36, …
  2. 8Next three: 0, 7, 26, 63, … and state the rule.
Show answers (try them yourself first)
  1. 115, 18, 21.
  2. 29, 11, 13.
  3. 326, 28, 30.
  4. 432, 64, 128.
  5. 581, 243, 729.
  6. 628, 36, 45.
  7. 749, 64, 81.
  8. 8124, 215, 342; each number is a cube minus 1.

Fractals

From daily life

A fern leaf, a head of broccoli and a winding river look rough and broken, not like circles or squares. A special geometry called fractals describes such natural shapes.

Simple definition

Fractals (फ्रॅक्टल्स) are shapes in which a small part, however small, looks like the whole figure. This is called self-similar (स्वसदृश).

Rule 1: Fractals in nature and the fractal triangle — Regular shapes (equal sides and angles) are easy to draw, but rough surfaces of planets, clouds, jagged seashores, rivers, mountains, forests, tree branches, broccoli and pineapples need the natural geometry of fractals. The Latin word means "to break". Mathematician Benoit Mandelbrot coined the term in 1975. Fractals appear in branching, geometric, spiral and algebraic forms. To draw the fractal triangle, take an equilateral triangle, join the midpoints of its sides to get four equal triangles, and repeat on the three corner triangles but not on the middle one.

1234

In the figure : In the figure joining the midpoints of the sides divides the triangle into four equal triangles. The process is repeated on the three gold corner triangles, but not on the middle triangle (4).

  • Easy : Easy: Ferns, clouds and rivers are natural shapes that a fractal can describe.
  • Medium : Medium: The word "fractal" comes from a Latin word meaning "to break".
  • Exam-type : Exam: Joining midpoints splits the triangle into 4 equal triangles, each with a quarter of the original area.

Rule 2: Counting the stages — Each stage splits every gold triangle into three new gold triangles. Stages 1, 2 and 3 have 3, 9 and 27 gold triangles. A small part of the fractal looks like the whole figure.

Stage 1Stage 2Stage 3

In the figure : In the figure stages 1, 2 and 3 have 3, 9 and 27 gold triangles. A small part of the fractal looks like the whole figure (self-similar).

  • Easy : Easy: Stage 1 has 3 gold triangles.
  • Medium : Medium: Stage 3 has 3 × 3 × 3 = 27 gold triangles.
  • Exam-type : Exam: Stage 4 has 27 × 3 = 81 gold triangles.
Remember

Self-similar. Coined by Mandelbrot in 1975. Triangles: 3, 9, 27, 81, … (× 3).

Common mistake

Repeating the process on the middle triangle too. Only the three corner triangles are repeated.

Shortcut

Memory clue: "भाग तोच, आकार तोच" (a small part looks like the whole).

Spot the difference

StageGold triangles
13
29
327
481

Practice questions

Easy
  1. 1Who coined the term fractal?
  2. 2In which year?
  3. 3What does the Latin word behind "fractal" mean?
Medium
  1. 4Name two natural things that look like fractals.
  2. 5Joining midpoints of a triangle's sides gives how many triangles?
  3. 6Which triangle is not repeated in the fractal triangle?
Challenging
  1. 7How many gold triangles are in stage 4?
  2. 8What does self-similar mean?
Show answers (try them yourself first)
  1. 1Benoit Mandelbrot.
  2. 21975.
  3. 3To break.
  4. 4Broccoli and fern leaves (also clouds, rivers).
  5. 54.
  6. 6The middle one.
  7. 781.
  8. 8A small part of the figure looks like the whole figure.

Quick revision

  • 1 is neither prime nor composite; 2 is the only even prime; twin primes differ by 2.
  • Triangular: 1, 3, 6, 10, 15; square: 1, 4, 9, 16, 25; cube: 1, 8, 27, 64, 125.
  • Fractals are self-similar; the term was coined by Benoit Mandelbrot in 1975.