RS Aggarwal Solutions for Class 6 Chapter 3 Whole Numbers

Question 1. Write down the smallest natural number.
Answer: The smallest natural number is 1.

Exam Tip: Remember that natural numbers start from 1, not from 0. This is a fundamental distinction to keep in mind.

Question 2. Write down the smallest whole number.
Answer: The smallest whole number is 0 (zero).

Exam Tip: Whole numbers include 0, which is the key difference from natural numbers. Always ensure you distinguish between the two sets.

Question 3. Write down, if possible, the largest natural number.
Answer: Every natural number has a successor that follows it. Therefore, there is no largest natural number.

Exam Tip: This is a key property to demonstrate - natural numbers are infinite, and you can always add 1 to get a bigger one.

Question 4. Write down, if possible, the largest whole number.
Answer: Every whole number has a successor that comes after it. Therefore, there is no largest whole number.

Exam Tip: Like natural numbers, whole numbers are also infinite. Emphasize that you can always find a larger whole number.

Question 5. Are all natural numbers also whole numbers?
Answer: Yes, all natural numbers are whole numbers.

Exam Tip: This is a directional relationship: every natural number belongs to the set of whole numbers, but the reverse is not true.

Question 6. Are all whole numbers also natural numbers?
Answer: No, all whole numbers are not natural numbers because 0 is a whole number but not a natural number.

Exam Tip: Use 0 as your counterexample - it is the distinguishing element between the two sets.

Question 7. Give successor of each of the whole numbers?
(i) 1000909
(ii) 2340900
(iii) 7039999
Answer:

Exam Tip: The successor is always found by adding 1 to the given number. Pay close attention when carrying over digits to avoid calculation errors.

Question 8. Write down the predecessor of each of the following whole numbers:
(i) 10000
(ii) 807000
(iii) 7005000
Answer:

Exam Tip: The predecessor is found by subtracting 1 from the given number. Watch for borrowing when dealing with zeros in the number.

Question 9. Represent the following numbers on the number line:
2, 0, 3, 5, 7, 11, 15
Answer:

Exam Tip: Mark each number with a dot or circle above the number line. Always ensure that the numbers are plotted in their correct positions according to their values.

Question 10. How many whole numbers are there between 21 and 61?
Answer: The whole numbers between 21 and 61 are 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59 and 60. Therefore, there are 39 whole numbers between 21 and 61.

Exam Tip: When finding numbers between two given values, exclude both endpoints. The difference 61 - 21 - 1 = 39 gives you the count directly.

Question 11. Fill in the blanks with the appropriate symbol < or >:
(i) 25...205
(ii) 170...107
(iii) 415...514
(iv) 10001...100001
(v) 2300014...2300041
Answer:
(i) 25 < 205
(ii) 170 > 107
(iii) 415 < 514
(iv) 10001 < 100001
(v) 2300014 < 2300041

Exam Tip: Compare numbers digit by digit from left to right. The smaller number always goes on the side where the symbol opens up (the symbol points to the smaller number).

Question 12. Arrange the following numbers is descending order:
925, 786, 1100, 141, 325, 886, 0, 270
Answer: Numbers in descending order: 1100, 925, 886, 786, 325, 270, 141, 0

Exam Tip: Descending order means from largest to smallest. Organize numbers by their magnitude - start with the largest and work your way down to the smallest.

Question 13. Write the largest number of 6 digits and the smallest number of 7 digits. Which one of these two is larger and by how much?
Answer: The largest six-digit number is 999,999. The smallest seven-digit number is 1,000,000. The smallest seven-digit number is larger than the largest six-digit number. The difference between these two numbers is 1,000,000 - 999,999 = 1. Therefore, the smallest seven-digit number is larger than the largest six-digit number by 1.

Exam Tip: Recognize that when you move from a 6-digit to a 7-digit number, the smallest 7-digit number is just 1 more than the largest 6-digit number. This demonstrates how place value works in our number system.

Question 14. Write down three consecutive whole numbers just preceding 8510001.
Answer: The first number before 8,510,001 is 8,510,001 - 1 = 8,510,000. The second number is 8,510,000 - 1 = 8,509,999. The third number is 8,509,999 - 1 = 8,509,998. Therefore, the three consecutive whole numbers that come just before 8,510,001 are 8,510,000, 8,509,999 and 8,509,998.

Exam Tip: Work backwards systematically - subtract 1 each time to find the previous whole number. List them in the order you found them to avoid confusion.

Question 15. Write down the next three consecutive whole numbers starting from 4009998.
Answer: Starting from 4,009,998, the first number is 4,009,998 + 1 = 4,009,999. The second number is 4,009,999 + 1 = 4,010,000. The third number is 4,010,000 + 1 = 4,010,001. Therefore, the next three consecutive whole numbers starting from 4,009,998 will be 4,009,999, 4,010,000 and 4,010,001.

Exam Tip: Work forwards methodically - add 1 each time to find the next whole number. Be careful with carrying when transitioning across zeros.

Question 16. Give arguments in support of the statement that there does not exist the largest natural number.
Answer: Each and every natural number has a successor that comes right after it. As a result, the largest natural number cannot exist.

Exam Tip: This is a proof by property - natural numbers are defined such that for any natural number n, n+1 is also a natural number. Use this defining property in your answer.

Question 17. Which of the following statements are true and which are false?
(i) Every whole number has its successor.
(ii) Every whole number has its predecessor.
(iii) 0 is the smallest natural number.
(iv) 1 is the smallest whole number.
(v) 0 is less than every natural number.
(vi) Between any two whole numbers there is a whole number.
(vii) Between any two non-consecutive whole numbers there is a whole number.
(viii) The smallest 5-digit number is the successor of the largest 4 digit number
(ix) Of the given two natural numbers, the one having more digits is greater.
(x) The predecessor of a two digit number cannot be a single digit number.
(xi) If a and b are natural numbers and a < b, than there is a natural number c such that a<b<c.
(xii) If a and b are whole numbers and a<b, then a+1< b+1.
(xiii) The whole number 1 has 0 as predecessor.
(xiv) The natural number 1 has no predecessor.
Answer:
(i) True - Every whole number has a successor that can be obtained by adding 1.
(ii) False - Zero (0) is a whole number whose predecessor (-1) is not a whole number.
(iii) False - 1 is the smallest natural number.
(iv) False - Zero (0) is the smallest whole number.
(v) True - The smallest natural number is 1, so zero (0) is less than every natural number.
(vi) False - No whole number exists between two consecutive whole numbers.
(vii) True - Between any two non-consecutive whole numbers there are one or more whole numbers.
(viii) True - The smallest five-digit number = 10,000 and the largest four-digit number = 9,999. The difference = 10,000 - 9,999 = 1. Because the difference is 1, 10,000 is the successor of 9,999.
(ix) True - When comparing two natural numbers with different numbers of digits, the one with more digits is always greater.
(x) False - 10 is a two-digit number whose predecessor is 9, which is a one-digit number.
(xi) False - If a and b are consecutive natural numbers, then there cannot be any natural number c in between a and b.
(xii) True
(xiii) True
(xiv) True - The predecessor of natural number 1 is 0, which is not a natural number.

Exam Tip: For true/false statements, always provide a brief reason or counterexample. This demonstrates your understanding and earns full marks in reasoning-based questions.

Question 18. The smallest natural number is
(a) 0
(b) 1
(c) -1
(d) None of these
Answer: (b) 1

Exam Tip: Natural numbers form a set that starts at 1 and increases indefinitely. Zero is not included in natural numbers by standard definition.

Question 19. The smallest whole number is
(a) 1
(b) 0
(c) -1
(d) None of these
Answer: (b) 0

Exam Tip: Whole numbers include 0 as their smallest element, making this the key distinction from natural numbers.

Question 20. The predecessor of 1 in natural numbers is
(a) 0
(b) 2
(c) -1
(d) None of these
Answer: (d) None of these
Since the smallest natural number is 1, its predecessor does not exist within the natural number set.

Exam Tip: Remember that 0, though it is the predecessor of 1 mathematically, is not a natural number. So within the natural numbers system, 1 has no predecessor.

Question 21. The predecessor of 1 in whole numbers is
(a) 0
(b) -1
(c) 2
(d) None of these
Answer: (a) 0
The predecessor of 1 = 1 - 1 = 0

Exam Tip: In whole numbers, 0 is included, so the predecessor of 1 is 0. This is different from natural numbers where the predecessor would not exist.

Question 22. The predecessor of 1 million is
(a) 9999
(b) 99999
(c) 999999
(d) 1000001
Answer: (c) 9,99,999
1 million = 10,00,000
Predecessor of 1 million = 10,00,000 - 1 = 9,99,999

Exam Tip: Break down large numbers into their place values to avoid errors when subtracting 1, especially when dealing with zeros.

Question 23. The successor of 1 million is
(a) 10001
(b) 100001
(c) 1000001
(d) 10000001
Answer: (c) 10,00,001
1 million = 10,00,000
Successor of 1 million = 10,00,000 + 1 = 10,00,001

Exam Tip: The successor is simply the number plus 1. Write large numbers with commas or in standard form to keep track of place values accurately.

Question 24. The product of the successor and predecessor of 99 is
(a) 9800
(b) 9900
(c) 1099
(d) 9700
Answer: (a) 9800
Successor of 99 = 99 + 1 = 100
Predecessor of 99 = 99 - 1 = 98
Their product = 100 × 98 = 9800

Exam Tip: Find successor and predecessor separately before multiplying. Recognize that 100 × 98 = (100)(100 - 2) = 10,000 - 200 = 9,800 for quick mental calculation.

Question 25. The product of a whole number (other than zero) and its successor is
(a) an even number
(b) an odd number
(c) divisible by 4
(d) divisible by 3
Answer: (a) an even number
Example: Whole number = 1, Successor of 1 = 1 + 1 = 2, Their product = 1 × 2 = 2. Thus, 2 is an even number.

Exam Tip: A whole number and its successor are always consecutive, meaning one is even and one is odd. The product of an even and odd number is always even.

Question 26. The product of the predecessor and successor of an odd natural number is always divisible by
(a) 2
(b) 4
(c) 6
(d) 8
Answer: (d) 8
The predecessor of an odd number is an even number. The successor of an odd number is also an even number. These two even numbers are two consecutive even numbers, and the product of two consecutive even numbers is always divisible by 8.

Exam Tip: When an odd number is flanked by its predecessor and successor, both neighbors are consecutive even numbers. Their product contains factors that guarantee divisibility by 8.

Question 27. The product of the predecessor and successor of an even natural number is
(a) divisible by 2
(b) divisible by 3
(c) divisible by 4
(d) an odd number
Answer: (d) an odd number
Example: Even number = 4, Predecessor of 4 = 4 - 1 = 3, Successor of 4 = 4 + 1 = 5, Their product = 3 × 5 = 15, which is an odd number.

Exam Tip: The predecessor and successor of an even number are both odd numbers. The product of two odd numbers is always odd, making this a reliable property.

Question 11. The successor of the smallest prime number is
(a) 1
(b) 2
(c) 3
(d) 4
Answer: (c) 3
In simple words: The smallest prime number is 2. When you add 1 to it, you get 3, which is the next number after 2.

Exam Tip: Remember that 2 is the smallest and only even prime number - this fact appears frequently in MCQs about primes.

Question 12. If x and y are co-primes, then their LCM is
(a) 1
(b) x/y
(c) xy
(d) None of these
Answer: (c) xy
In simple words: When two numbers share no common factor except 1 (called co-primes), their least common multiple is simply the product of those two numbers.

Exam Tip: This is a key property of co-primes - always recall that LCM(a,b) = a × b when a and b are co-primes, and HCF(a,b) = 1.

Question 13. The HCF of two co-primes is
(a) the smaller number
(b) the larger number
(c) product of the numbers
(d) 1
Answer: (d) 1
In simple words: Co-primes are numbers that have no common factor other than 1, so their highest common factor must always be 1.

Exam Tip: Co-primes by definition have HCF = 1 - this is the defining characteristic that makes them co-primes in the first place.

Question 14. The smallest number which is neither prime nor composite is
(a) 0
(b) 1
(c) 2
(d) 3
Answer: (b) 1
In simple words: The number 1 stands alone - it is not considered prime (since primes have exactly two factors) and not composite (since composites have more than two factors).

Exam Tip: Always remember that 1 is a unique number with a special status - it is neither prime nor composite by definition.

Question 15. The product of any natural number and the smallest prime is
(a) an even number
(b) an odd number
(c) a prime number
(d) None of these
Answer: (a) an even number
In simple words: The smallest prime number is 2. Whenever you multiply any natural number by 2, the result is always even.

Exam Tip: Multiplying any number by 2 automatically makes it even - this follows directly from the definition of even numbers as multiples of 2.

Question 16. Every counting number has an infinite number of
(a) factors
(b) multiples
(c) prime factors
(d) None of these
Answer: (b) multiples
In simple words: You can keep multiplying any number by 1, 2, 3, 4... forever, creating endless multiples. However, factors are limited - each number has only a fixed set of factors.

Exam Tip: Distinguish clearly: factors are finite and divide the number, while multiples are infinite and the number divides them.

Question 17. The product of two numbers is 1530 and their HCF is 15. The LCM of these numbers is
(a) 102
(b) 120
(c) 84
(d) 112
Answer: (a) 102
In simple words: There is a useful relationship: when you multiply the HCF and LCM of two numbers, you get their product. So LCM = Product ÷ HCF = 1530 ÷ 15 = 102.

Exam Tip: Always remember the formula: Product of two numbers = HCF × LCM - use it to find any missing value.

Question 18. The least number divisible by each of the numbers 15, 20, 24 and 32 is
(a) 960
(b) 480
(c) 360
(d) 640
Answer: (b) 480
In simple words: To find the least number divisible by all four numbers, we find their LCM using prime factorization: 15 = 3 × 5, 20 = 2² × 5, 24 = 2³ × 3, 32 = 2⁵. The LCM takes the highest power of each prime: 2⁵ × 3 × 5 = 480.

Exam Tip: When finding LCM, always take the highest power of each prime factor that appears - this ensures divisibility by all given numbers.

Question 19. The greatest number which divides 134 and 167 leaving 2 as remainder in each case is
(a) 14
(b) 19
(c) 33
(d) 17
Answer: (c) 33
In simple words: Since both numbers leave a remainder of 2, we subtract 2 from each: 134 - 2 = 132 and 167 - 2 = 165. Then we find their HCF. Breaking them down: 132 = 2² × 3 × 11 and 165 = 3 × 5 × 11. Their HCF = 3 × 11 = 33.

Exam Tip: When a number divides two values with the same remainder, first subtract that remainder, then find the HCF of the results.

Question 20. Which of the following numbers is a prime number?
(a) 91
(b) 81
(c) 87
(d) 97
Answer: (d) 97
In simple words: Check each option: 91 = 7 × 13, 81 = 3⁴, 87 = 3 × 29. All of these have factors other than 1 and themselves. However, 97 cannot be divided by any number except 1 and 97, making it prime.

Exam Tip: To verify if a number is prime, test divisibility by small primes up to its square root - if none divide it, the number is prime.

Question 21. If two numbers are equal, then
(a) their LCM is equal to their HCF
(b) their LCM is less than their HCF
(c) their LCM is equal to two times their HCF
(d) None of these
Answer: (a) their LCM is equal to their HCF
In simple words: When two numbers are identical, both their LCM and HCF are that same number itself - they are always equal to each other.

Exam Tip: When a = b, then LCM(a,b) = HCF(a,b) = a = b - this is a straightforward but important special case.

Question 22. a and b are two co-primes. Which of the following is/are true?
(a) LCM (a, b) = a × b
(b) HCF (a, b) = 1
(c) Both (a) and (b)
(d) Neither (a) nor (b)
Answer: (c) Both (a) and (b)
In simple words: Co-primes have no common factor except 1, so their HCF is always 1. Also, for co-primes, the LCM equals their product. Both statements are true.

Exam Tip: Remember the two defining properties of co-primes: HCF = 1 and LCM = product - these go hand in hand.