MATHS 2 Marks Questions

We need to write factors of 10000.
10000= 10 × 10 × 10 × 10
= 2 × 5 × 2 × 5 × 2 × 5 × 2 × 5
So, 2 × 2 × 2 × 2 = 16 and 5 × 5 × 5 × 5 = 625.
Hence, 16 and 625 are the two numbers whose product is 10000.

If a number is divisible by 8, it will automatically be divisible by 4.
If a number is divisible by 10, it is also divisible by 2 and 5. Therefore, checking divisibility by 8 and 10 confirms divisibility by all other numbers (2, 4, 5).
Thus, the pair of numbers that Guna could check to determine that 14560 is divisible by all of 2, 4, 5, 8, and 10 is: 8 and 5.

(a) Sometimes true. Sum of any two even numbers is not always divisible by 4. For e×ample, 6 + 4 = 10 which is not divisible by 4 whereas 2+2 = 4 which is divisible of 4.
(b) Sometimes true. Sum of two odd numbers can indeed be even but not necessarily a multiple of 4. For e×ample, 1 + 5 = 6 which is not a multiple of 4 whereas 1+3 = 4, which is a multiple of 4. Similarly 7 + 5 = 12, which is a multiple of 4.

(i) Let the year you were born be 2010.
2010 (not divisible by 4) 
2011 (not divisible by 4) 
2012 (divisible by 4) 
2013 (not divisible by 4) 
2014 (not divisible by 4) 
2015 (not divisible by 4) 
2016 (divisible by 4) 
2017 (not divisible by 4) 
2018 (not divisible by 4) 
2019 (not divisible by 4) 
2020 (divisible by 4) 
2021 (not divisible by 4) 
2022 (not divisible by 4) 
2023 (not divisible by 4) 
2024 (divisible by 4)
Thus, the years that were leap years are 2012, 2016, 2020 and 2024.
(ii) Since, leap years occur in the years that are multiples of 4, except for those years that are evenly divisible by 100 but not 400. Therefore, the leap years from 2024 till 2099 are 2024, 2028, 2032, 2036, 2040, 2044, 2048, 2052, 2056, 2060, 2064, 2068, 2072, 2076, 2080, 2084, 2088, 2092 and 2096.
Hence, there are 19 leap years from the year 2024 till 2099.

(a) The smallest prime numbers are 2, 3, and 5. To find the smallest number with these primes as factors, multiply them together:
2 × 3 × 5 = 30
So, the smallest number whose prime factorization has three different prime numbers is 30.
(b) The smallest four prime numbers are 2, 3, 5, and 7. To find the smallest number with these primes as factors, multiply them together:
2 × 3 × 5 × 7 = 210
Thus, the smallest number whose prime factorization has four different prime numbers is 210.

The prime factorization of 1955:
1955 = 5 × 17 × 23
All the factors are prime numbers and are less than 30.
Hence, the three prime numbers whose product is 1955 are 5, 17, and 23.

To find the number, we multiply these prime factors together:
2 × 3 × 3 × 11 = 198
Thus, the number is 198.

The five prime numbers for which doubling and adding 1 gives another prime are:
• 2 (since 2 × 2 + 1 = 5)
• 3 (since 2 × 3 + 1 = 7)
• 5 (since 2 × 5 + 1 = 11)
• 11 (since 2 × 11 + 1 = 23)
• 23 (since 2 × 23 + 1 = 47)

2, 4 and 5 cannot form a single prime number.
Because, when its units digit is 2 or 4 it is divided by 2, and when units digits is 5 it is divided by 5 so that’s why 2, 4 and 5 cannot form a prime number.

Here, 45 = 3 × 3 × 5 (2 distinct primes)
60 = 2 × 2 × 3 × 5(3 distinct primes)
91 = 7 × 13 (2 distinct primes)
105 = 3 × 5 × 7 (3 distinct primes)
330 = 2 × 3 × 5 × 11 (4 distinct primes)
Number 105 is the product of exactly three distinct prime numbers i.e. 3 × 5 × 7.

No, 2 is the only even prime number. Since 2 is the only even number that meets the criteria of a prime number (its only divisors are 1 and 2), it is the only even prime number. All other even numbers are divisible by 2 and at least one other number, so they are not prime.

There is not an equal number of primes in every row. The number of primes varies between rows. The decade 90-99 has the least number of primes with only 1 prime (97).
The decades 0-9 and 10-19 have the greatest number of primes, each with 4 primes.

Multiples of 6 Multiples of 8

Here 6 could also be replaced by 3.
As 24, 48, 72, are also common multiples of 3 and 8.

Factors of 28 = 1, 2, 4, 7, 14, 28
Factors of 70 = 1,2, 5, 7, 10, 14, 35, 70
Common factors are 1, 2, 7 and 14
Hence jump sizes which will land at both 28 and 70 are 1, 2, 7 and 14.

If ‘idli-vada’ is said after number 50 it means that the least common multiple (LCM) of the two numbers must be slightly greater than 50. The LCM of 6 and 9 is 54,’.which is the first common multiple after 50, making 6 and 9 the possible numbers. Hence the two numbers could be 6 and 9.

Numbers that are multiples of 25 are 25, 50, 75, 100, 125, 150, 175, …
Numbers that are multiples of 50 are 50, 100, 150, 200, 250,300,…
Hence, the numbers that are multiples of 25 but not multiples of 50 are 25, 75, 125, 175,…

The only perfect number between 1 and 10 is 6.
• Proper divisors are 1, 2, 3, 6
• Sum of proper divisors: 1 + 2 + 3 + 6 = 12
• 12 is twice of 6, hence 6 is a perfect number.

(a) 7 is the common factors of 7, 14, 21,28, 35, which are less than 40. And there is one number which have digit sum of 8, is 35 = (3 + 5) = 8.
So, I am 35.
(b) Common factors of 3 and 5 are 15, 30, 45, 60, 75,90, (which are less than 100). And there is one number which one of digit is 1 more than the other that is 45. So, I am 45.

Yes, this figure is related to the ‘idli-vada’ game. Figure below for game played till 60.

To determine the 10th occurrence of “idli- vada”; we need to identify the numbers that are multiples of both 3 and 5.
The numbers for which “idli-vada” is said are the multiples of 15.
This sequence is: 15, 30, 45, 60, 75, 90, 105, 120, 135, 150,…
Thus, the 10th number for which players should say “idli-vada” is 150.

The numbers between 330 and 340 that are divisible by 4 are 332, 336, and 340. The numbers between 1730 and 1740 that are divisible by 4 are 1732, 1736, and 1740. The numbers between 2030 and 2040 that are divisible by 4 are 2032, 2036, and 2040.

(a) Factors of 20 are 1, 2, 4, 5, 10, 20 Factors of 28 are 1, 2, 4, 7, 14, 28 Common factors are 1, 2, 4
(b) Factors of 35 are 1, 5, 7, 35 Factors of 50 are 1, 2,’5, 10, 25, 50 Common factors are 1,5
(c) Factors of 4 are 1, 2, 4 Factors of 8 are 1, 2, 4, 8 Factors of 12 are 1, 2, 3, 4, 6, 12 Common factors are 1, 2, 4
(d) Factors of 5 are 1, 5 Factors of 15 are 1, 3, 5, 15 Factors of 25 are 1, 5, 25 Common factors are 1,5.

There are 300 multiples of 3 between 1 and 900 and there are 180 multiples of 5 between 1 and 900. There are 60 multiples of 15 between 1 and 900.
(a) “idli” is said: 300 times (including the times “idli-vada” is said).
(b) “vada” is said: 180 times (including the times “idli-vada” is said).
(c) “idli-vada” is said: 60 times.

(a) 23 = 1 × 23
Hence, 23 is a prime number.

(b) Since 18 = 1 × 18 or 6 × 3
Hence, 18 is not a prime number.

(c) Since 25 = 1 × 25 or 5× 5
Hence, 25 is not a prime number.

(d) Since 15 = 1 × 15 or 3 × 5
Hence, 15 is not a prime number.
Hence, the option (a) is correct.

(a) Given numbers are: 4, 8 and 12
Factors of 4 are 1,2, 4
Factors of 8 are 1, 2, 4, 8
Factors of 12 are 1, 2, 3, 4, 6, 12
Therefore, the common factors of 4, 8 and 12 are 1,2, and 4.

(b) Given numbers are: 5, 15 and 25
Factors of 5 are 1, 5
Factors of 15 are 1, 3, 5, 15
Factors of 25 are 1, 5, 25
Therefore, the common factors of 5, 15, and 25 are 1 and 5.

The major differences between factors and multiples are provided below:

Factors	Multiples
A factor of a number is defined as an exact divisor of the given number.	A multiple of a number is defined as a number that is obtained by multiplying it by a natural number.
For example, the factors of 20 are 1, 2, 4, 5, 10, and 20.	For example, the multiples of 20 are 20, 40, 60, 80, 100, etc.