The World of Numbers | Exercise 3.1

Question 2

Look at the sequence of numbers on one column of the Ishango bone: 11,13,17,1911, 13, 17, 19. What do these numbers have in common? List the next three numbers that fit this pattern.

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Solution
Understand the Question
  • A prime number is a natural number greater than 11 that has no positive divisors other than 11 and itself.
  • The given numbers 11,13,17,1911, 13, 17, 19 are all consecutive prime numbers.
  • To find the next three numbers fitting this pattern, we find the next three prime numbers greater than 1919.

Step 1 · Identify the Common Property

Check the factors of each given number:

Factors of 11:1,11Factors of 13:1,13Factors of 17:1,17Factors of 19:1,19\begin{aligned} \text{Factors of } 11 &: 1, 11 \\ \text{Factors of } 13 &: 1, 13 \\ \text{Factors of } 17 &: 1, 17 \\ \text{Factors of } 19 &: 1, 19 \end{aligned}

Each number has exactly two distinct factors (11 and itself). Therefore, all given numbers are prime numbers.

Step 2 · Find the Next Three Prime Numbers

Check the numbers following 1919 to find the next three primes:

20not prime (divisible by 2,4,5,10)21not prime (divisible by 3,7)22not prime (divisible by 2,11)23prime (divisible only by 1,23)24not prime25not prime26not prime27not prime28not prime29prime (divisible only by 1,29)30not prime31prime (divisible only by 1,31)\begin{aligned} 20 &\quad\text{not prime (divisible by } 2, 4, 5, 10) \\ 21 &\quad\text{not prime (divisible by } 3, 7) \\ 22 &\quad\text{not prime (divisible by } 2, 11) \\ 23 &\quad\text{prime (divisible only by } 1, 23) \\ 24 &\quad\text{not prime} \\ 25 &\quad\text{not prime} \\ 26 &\quad\text{not prime} \\ 27 &\quad\text{not prime} \\ 28 &\quad\text{not prime} \\ 29 &\quad\text{prime (divisible only by } 1, 29) \\ 30 &\quad\text{not prime} \\ 31 &\quad\text{prime (divisible only by } 1, 31) \end{aligned}

The next three prime numbers are 2323, 2929, and 3131.

Answer

The numbers are all prime numbers, and the next three numbers are 23,29,3123, 29, 31.

Common Mistakes
  • Confusing Odd Numbers with Prime Numbers: Mistaking the sequence as an odd-number pattern and including composite odd numbers like 2121, 2525, or 2727.
  • Missing Consecutive Primes: Skipping a prime number such as 2323 or 2929 while listing subsequent values.

More questions in Exercise 3.1

Q1

A merchant in the port city of Lothal is exchanging bags of spices for copper ingots. He receives 15 ingots for every 2 bags of spices. If he brings 12 bags of spices to the market, how many copper ingots will he leave with?

Q2

Look at the sequence of numbers on one column of the Ishango bone: 11,13,17,1911, 13, 17, 19. What do these numbers have in common? List the next three numbers that fit this pattern.

Q3

We know that Natural Numbers are closed under addition (the sum of any two natural numbers is always a natural number). Are they closed under subtraction? Provide a couple of examples to justify your answer.

Q4

Ancient Indians used the joints of their fingers to count, a practice still seen today. Each finger has 3 joints, and the thumb is used to count them. How many can you count on one hand? How does this relate to the ancient base-12 counting systems?

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