Prime Time | FIO

Question 23

Identify whether each statement is true or false. Explain.

a. There is no prime number whose units digit is 4.

b. A product of primes can also be prime.

c. Prime numbers do not have any factors.

d. All even numbers are composite numbers.

e. 2 is a prime and so is the next number, 3. For every other prime, the next number is composite.

Solution

Understand the Question

A prime number is a natural number greater than $1$ that has exactly two distinct factors: $1$ and the number itself (e.g., $2, 3, 5, 7, 11$ ).
A composite number has more than two factors (e.g., $4, 6, 8, 9$ ).
The number $2$ is the smallest prime and the only even prime number.

(a) There is no prime number whose units digit is 4.

Step 1 · Analyze Numbers Ending in 4

Any number whose units digit is $4$ (such as $4, 14, 24, 34, \dots$ ) is an even number and is divisible by $2$ .

This means any number ending in $4$ has at least three factors: $1$ , $2$ , and the number itself.

Since a prime number must have exactly two factors and the only even prime number is $2$ (which does not end in $4$ ), no prime number can have a units digit of $4$ .

Answer

(a) True

(b) A product of primes can also be prime.

Step 1 · Evaluate Product of Primes

Consider the product of two prime numbers, $2$ and $3$

\begin{aligned} 2 \times 3 &= 6 \end{aligned}

The factors of $6$ are $1, 2, 3,$ and $6$ . Because it has more than two factors, $6$ is composite.

The product of two or more prime numbers will always have the primes themselves as factors in addition to $1$ and the product, making the result composite.

Answer

(b) False

(c) Prime numbers do not have any factors.

Step 1 · Examine Factors of Prime Numbers

By definition, a prime number is a number greater than $1$ that has exactly two factors: $1$ and the number itself.

For example:

Factors of $7$ are $1$ and $7$ .
Factors of $11$ are $1$ and $11$ .

Therefore, prime numbers do have factors.

Answer

(c) False

(d) All even numbers are composite numbers.

Step 1 · Check the Even Number 2

An even number is any integer divisible by $2$ .

Consider the number $2$ :

$2$ is an even number.
Its only factors are $1$ and $2$ .

Since $2$ has exactly two factors, it is a prime number, not composite. Therefore, not all even numbers are composite.

Answer

(d) False

(e) 2 is a prime and so is the next number, 3. For every other prime, the next number is composite.

Step 1 · Analyze Subsequent Numbers for Other Primes

$2$ and $3$ are both prime numbers.

Every prime number greater than $3$ (such as $5, 7, 11, 13, \dots$ ) is an odd number.

The number immediately following any odd number is always an even number greater than $2$ (e.g., after $5$ comes $6$ ; after $7$ comes $8$ ).

Since all even numbers greater than $2$ are composite, the next number after any prime greater than $3$ is always composite.

Answer

(e) True

Common Mistakes

Overlooking 2 as a Prime: Forgetting that $2$ is an even number that is prime, making statements like "all even numbers are composite" false.
Misunderstanding "No Factors": Confusing having no other factors except 1 and itself with having no factors at all.

More questions in FIO

At what number is 'idli-vada' said for the 10th time?

If the game is played for the numbers 1 to 90, find out:

a. How many times would the children say 'idli' (including the times they say 'idli-vada')? b. How many times would the children say 'vada' (including the times they say 'idli-vada')? c. How many times would the children say 'idli-vada'?

What if the game was played till 900? How would your answers change?

Is this figure somehow related to the 'idli-vada' game?

Hint: Imagine playing the game till 30. Draw the figure if the game is played till 60.

Find all multiples of 40 that lie between 310 and 410.

Who am I?

a. I am a number less than 40. One of my factors is 7. The sum of my digits is 8.

b. I am a number less than 100. Two of my factors are 3 and 5. One of my digits is 1 more than the other.

A number for which the sum of all its factors is equal to twice the number is called a perfect number. The number 28 is a perfect number. Its factors are 1, 2, 4, 7, 14 and 28. Their sum is 56 which is twice 28. Find a perfect number between 1 and 10.

Find the common factors of:

a. 20 and 28

b. 35 and 50

c. 4, 8 and 12

d. 5, 15 and 25

Find any three numbers that are multiples of 25 but not multiples of 50.

Q10

Anshu and his friends play the 'idli-vada' game with two numbers, which are both smaller than 10. The first time anybody says 'idli-vada' is after the number 50. What could the two numbers be which are assigned 'idli' and 'vada'?

Q11

In the treasure hunting game, Grumpy has kept treasures on 28 and 70. What jump sizes will land on both the numbers?

Q12

In the diagram below, Guna has erased all the numbers except the common multiples. Find out what those numbers could be and fill in the missing numbers in the empty regions.

Q13

Find the smallest number that is a multiple of all the numbers from 1 to 10, except for 7.

Q14

Find the smallest number that is a multiple of all the numbers from 1 to 10.

Q15

We see that $2$ is a prime and also an even number. Is there any other even prime?

Q16

Look at the list of primes till 100. What is the smallest difference between two successive primes? What is the largest difference?

Q17

Are there an equal number of primes occurring in every row in the table on the previous page? Which decades have the least number of primes? Which have the most number of primes?

Q18

Which of the following numbers are prime: 23, 51, 37, 26?

Q19

Write three pairs of prime numbers less than 20 whose sum is a multiple of 5.

Q20

The numbers 13 and 31 are prime numbers. Both these numbers have same digits 1 and 3. Find such pairs of prime numbers up to 100.

Q21

Find seven consecutive composite numbers between 1 and 100.

Q22

Twin primes are pairs of primes having a difference of $2$ . For example, $3$ and $5$ are twin primes. So are $17$ and $19$ . Find the other twin primes between $1$ and $100$ .

Q23

Identify whether each statement is true or false. Explain.

a. There is no prime number whose units digit is 4.

b. A product of primes can also be prime.

c. Prime numbers do not have any factors.

d. All even numbers are composite numbers.

e. 2 is a prime and so is the next number, 3. For every other prime, the next number is composite.

Q24

Which of the following numbers is the product of exactly three distinct prime numbers: 45, 60, 91, 105, 330?

Q25

How many three-digit prime numbers can you make using each of 2, 4 and 5 once?

Q26

Observe that 3 is a prime number, and $2 \times 3 + 1 = 7$ is also a prime. Are there other primes for which doubling and adding 1 gives another prime? Find at least five such examples.

Q27

Find the prime factorisations of the following numbers: 64, 104, 105, 243, 320, 141, 1728, 729, 1024, 1331, 1000.

Q28

The prime factorisation of a number has one 2, two 3s, and one 11. What is the number?

Q29

Find three prime numbers, all less than 30, whose product is 1955.

Q30

Find the prime factorisation of these numbers without multiplying first

a. $56 \times 25$

b. $108 \times 75$

c. $1000 \times 81$

Q31

What is the smallest number whose prime factorisation has:

a. three different prime numbers?

b. four different prime numbers?

Q32

Are the following pairs of numbers co-prime? Guess first and then use prime factorisation to verify your answer.

a. 30 and 45

b. 57 and 85

c. 121 and 1331

d. 343 and 216

Q33

Is the first number divisible by the second? Use prime factorisation.

a. 225 and 27

b. 96 and 24

c. 343 and 17

d. 999 and 99

Q34

The first number has prime factorisation $2 \times 3 \times 7$ and the second number has prime factorisation $3 \times 7 \times 11$ . Are they co-prime? Does one of them divide the other?

Q35

Guna says, “Any two prime numbers are co-prime?”. Is he right?

Q36

2024 is a leap year (as February has 29 days). Leap years occur in the years that are multiples of 4, except for those years that are evenly divisible by 100 but not 400.

a. From the year you were born till now, which years were leap years? b. From the year 2024 till 2099, how many leap years are there?

Q37

Find the largest and smallest 4-digit numbers that are divisible by 4 and are also palindromes.

Q38

Explore and find out if each statement is always true, sometimes true or never true. You can give examples to support your reasoning.

a. Sum of two even numbers gives a multiple of 4. b. Sum of two odd numbers gives a multiple of 4.

Q39

Find the remainders obtained when each of the following numbers are divided by (a) 10, (b) 5, (c) 2.

78, 99, 173, 572, 980, 1111, 2345

Q40

The teacher asked if 14560 is divisible by all of 2, 4, 5, 8 and 10. Guna checked for divisibility of 14560 by only two of these numbers and then declared that it was also divisible by all of them. What could those two numbers be?

Q41

Which of the following numbers are divisible by all of 2, 4, 5, 8 and 10: 572, 2352, 5600, 6000, 77622160.

Q42

Write two numbers whose product is 10000. The two numbers should not have 0 as the units digit.