Prime Time | FIO

Question 36

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?

Solution

We will check each year using the special rules for leap years.

Step 1 — Finding leap years from 2010 to 2024

Leap years divide by 4 evenly. But there is a special rule. Years dividing by 100 are not leap years. Unless they also divide by 400. We will check years from 2010 to 2024.

Let us list the years: 2010: Does not divide by 4. 2011: Does not divide by 4. 2012: Divides by 4. It is not a multiple of 100. So, 2012 is a leap year. 2013: Does not divide by 4. 2014: Does not divide by 4. 2015: Does not divide by 4. 2016: Divides by 4. It is not a multiple of 100. So, 2016 is a leap year. 2017: Does not divide by 4. 2018: Does not divide by 4. 2019: Does not divide by 4. 2020: Divides by 4. It is not a multiple of 100. So, 2020 is a leap year. 2021: Does not divide by 4. 2022: Does not divide by 4. 2023: Does not divide by 4. 2024: Divides by 4. It is not a multiple of 100. So, 2024 is a leap year.

The leap years are 2012, 2016, 2020, and 2024. We count these leap years.

$\text{Number of leap years} = 4$

$\boxed{4 \text{ leap years}}$

Step 2 — Counting leap years from 2024 to 2099

We need to find all leap years in this range. First, we find years that divide by 4. The first year is 2024. It divides by 4. The last year is 2099. It does not divide by 4. The last year that divides by 4 before 2099 is 2096.

We count how many numbers divide by 4 from 2024 to 2096. We use a simple counting method.

$\text{Count} = (\text{Last year} - \text{First year}) \div 4 + 1$

$\text{Count} = (2096 - 2024) \div 4 + 1$

$\text{Count} = 72 \div 4 + 1$

$\text{Count} = 18 + 1$

$\text{Count} = 19$

Now we check the special rule. We look for years that divide by 100. But they must not divide by 400. In our range, 2024 to 2099, there are no years that divide by 100. The next year dividing by 100 after 2000 is 2100. So, all years that divide by 4 in this range are leap years.

$\boxed{19 \text{ leap years}}$

Answer

(a) The leap years from 2010 to 2024 are 2012, 2016, 2020, and 2024. There are 4 leap years. (b) The leap years from 2024 to 2099 are 19 in total.

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

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Q11

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Q12

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Q13

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Q14

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Q15

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Q16

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Q17

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Q18

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Q19

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Q20

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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

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Q25

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Q26

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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

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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

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Q41

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Q42

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