Look at the sequence of numbers on one column of the Ishango bone: 11, 13, 17, 19. What do these numbers have in common? List the next three numbers that fit this pattern.
Step-by-step solution
Idea: A prime number has exactly two factors, 1 and itself. Check each number, then keep testing the numbers after 19 one by one.
- 11, 13, 17 and 19 each have only two factors, 1 and the number itself. So they are all prime numbers; they are exactly the primes between 10 and 20.1 mark
- Test the numbers after 19: 20 = 2 × 10, 21 = 3 × 7, 22 = 2 × 11 are not prime; 23 is prime; 24, 25 = 5 × 5, 26, 27 = 3 × 9, 28 are not; 29 is prime; 30 is not; 31 is prime.½ mark
- The next three numbers are 23, 29, 31.½ mark
Check: 23, 29 and 31 are not divisible by 2, 3 or 5, and 5 × 5 = 25 < 31 < 36 = 6 × 6, so checking 2, 3 and 5 is enough ✓.
Answer to write in the exam
11, 13, 17, 19 each have only two factors, 1 and the number itself.
∴ They are prime numbers (the primes between 10 and 20).
20, 21, 22 not prime; 23 prime; 24–28 not prime; 29 prime; 30 not prime; 31 prime
∴ Next three numbers: 23, 29, 31
Common mistakes that cost marks
- Calling them ‘odd numbers’. They are odd, but 15 is odd too and is missing from the list; the pattern is primes.
- Listing 21 or 27 as the next numbers. 21 = 3 × 7 and 27 = 3 × 9 are not prime.
- Forgetting 29 and jumping from 23 to 31.
How this can come in the exam
How many prime numbers lie between 30 and 50?
- 4
- 5
- 6
- 7
Show answer
(B) 5
31, 37, 41, 43, 47: five primes.
Assertion (A): 51 is a prime number.
Reason (R): 51 is an odd number.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(D) A is false but R is true.
51 = 3 × 17, so A is false. R is true: 51 is odd. Being odd does not make a number prime.
Try one yourself
List all the prime numbers between 40 and 60.
Show answer
41, 43, 47, 53, 59.
More questions like this
- 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.
- 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?
- Brahmagupta did not stop at zero. He realised that if subtraction of a number from itself can result in zero (5 − 5 = 0), then what would happen if we subtracted a larger number from a smaller one (3 − 5 = □)?
- Why does a negative times a negative equal a positive? Think of it in terms of action and debt. If a negative number represents a debt, then multiplying by a negative number represents the removal of that debt.
(Hint: If someone takes away (−) four of your debts that are each worth ₹3 (that is, −3), you are effectively ₹12 richer! Therefore, (−3) × (−4) = +12.) - The temperature in the high-altitude desert of Ladakh is recorded as 4 °C at noon. By midnight, it drops by 15 °C. What is the midnight temperature?