Playing with Numbers-Solutions Ex. 3.3
CBSE Class –VI Mathematics
NCERT Solutions
Chapter 3 Playing With Numbers (Ex. 3.3)
NCERT Solutions
Chapter 3 Playing With Numbers (Ex. 3.3)
Question 1. Using divisibility test, determine which of the following numbers are divisible by 2; by 3; by 4; by 5; by 6; by 8; by 9; by 10; by 11. (say yes or no)
Number | Divisible by | ||||||||
2 | 3 | 4 | 5 | 6 | 8 | 9 | 10 | 11 | |
128 990 1586 275 6686 639210 429714 2856 3060 406839 | Yes | No | Yes | No | No | Yes | No | No | No |
Answer:
Number | Divisible by | ||||||||
2 | 3 | 4 | 5 | 6 | 8 | 9 | 10 | 11 | |
128 990 1586 275 6686 639210 429714 2856 3060 406839 | Yes Yes Yes No Yes Yes Yes Yes Yes No | No Yes No No No Yes Yes Yes Yes Yes | Yes No No No No No No Yes Yes No | No Yes No Yes No Yes No No Yes No | No Yes No No No Yes Yes Yes Yes No | Yes No No No No No No Yes No No | No Yes No No No No Yes No Yes No | No Yes No No No Yes No No Yes No | No Yes No Yes No Yes No No No No |
Question 2. Using divisibility test, determine which of the following numbers are divisibly by 4; by 8:
(a) 572
(b) 726352
(c) 550
(d) 6000
(e) 12159
(f) 14560
(g) 21084
(h) 31795072
(i) 1700
(j) 2150
(b) 726352
(c) 550
(d) 6000
(e) 12159
(f) 14560
(g) 21084
(h) 31795072
(i) 1700
(j) 2150
Answer:
(a) 572
Divisible by 4 as its last two digits are divisible by 4.
Divisible by 4 as its last two digits are divisible by 4.
Not divisible by 8 as its last three digits are not divisible by 8.
(b) 726352
Divisible by 4 as its last two digits are divisible by 4.
Divisible by 8 as its last three digits are divisible by 8.
(c) 5500
Divisible by 4 as its last two digits are divisible by 4.
Not divisible by 8 as its last three digits are not divisible by 8.
(d) 6000
Divisible by 4 as its last two digits are 0.
Divisible by 8 as its last three digits are 0.
(e) 12159
Not divisible by 4 and 8 as it is an odd number.
(f) 14560
Divisible by 4 as its last two digits are divisible by 4.
Divisible by 8 as its last three digits are divisible by 8.
(g) 21084
Divisible by 4 as its last two digits are divisible by 4.
Not divisible by 8 as its last three digits are not divisible by 8.
(h) 31795072
Divisible by 4 as its last two digits are divisible by 4.
Divisible by 8 as its last three digits are divisible by 8.
(i) 1700
Divisible by 4 as its last two digits are 0.
Not divisible by 8 as its last three digits are not divisible by 8.
(j) 5500
Divisible by 4 as its last two digits are 0.
Not divisible by 8 as its last three digits are not divisible by 8.
Question 3. Using divisibility test, determine which of the following numbers are divisible by 6:
(a) 297144
(b) 1258
(c) 4335
(d) 61233
(e) 901352
(f) 438750
(g) 1790184
(h) 12583
(i) 639210
(j) 17852
(b) 1258
(c) 4335
(d) 61233
(e) 901352
(f) 438750
(g) 1790184
(h) 12583
(i) 639210
(j) 17852
Answer:
(a) 297144
Divisible by 2 as its units place is an even number.
Divisible by 3 as sum of its digits (= 27) is divisible by 3.
Since the number is divisible by both 2 and 3, therefore, it is also divisible by 6.
(b) 1258
Divisible by 2 as its units place is an even number.
Not divisible by 3 as sum of its digits (= 16) is not divisible by 3.
Since the number is not divisible by both 2 and 3, therefore, it is not divisible by 6.
(c) 4335
Not divisible by 2 as its units place is not an even number.
Divisible by 3 as sum of its digits (= 15) is divisible by 3.
Since the number is not divisible by both 2 and 3, therefore, it is not divisible by 6.
(d) 61233
Not divisible by 2 as its units place is not an even number.
Divisible by 3 as sum of its digits (= 15) is divisible by 3.
Since the number is not divisible by both 2 and 3, therefore, it is not divisible by 6.
(e) 901352
Divisible by 2 as its units place is an even number.
Not divisible by 3 as sum of its digits (= 20) is not divisible by 3.
Since the number is not divisible by both 2 and 3, therefore, it is not divisible by 6.
(f) 438750
Divisible by 2 as its units place is 0.
Divisible by 3 as sum of its digits (= 27) is divisible by 3.
Since the number is divisible by both 2 and 3, therefore, it is divisible by 6.
(g) 1790184
Divisible by 2 as its units place is an even number.
Divisible by 3 as sum of its digits (= 30) is divisible by 3.
Since the number is divisible by both 2 and 3, therefore, it is divisible by 6.
(h) 12583
Not divisible by 2 as its units place is not an even number.
Not divisible by 3 as sum of its digits (= 19) is not divisible by 3.
Since the number is not divisible by both 2 and 3, therefore, it is not divisible by 6.
(i) 639210
Divisible by 2 as its units place is 0.
Divisible by 3 as sum of its digits (= 21) is divisible by 3.
Since the number is divisible by both 2 and 3, therefore, it is divisible by 6.
(j) 17852
Divisible by 2 as its units place is an even number.
Not divisible by 3 as sum of its digits (= 23) is not divisible by 3.
Since the number is not divisible by both 2 and 3, therefore, it is not divisible by 6.
Question 4. Using divisibility test, determine which of the following numbers are divisible by 11:
(a) 5445
(b) 10824
(c) 7138965
(d) 70169308
(e) 10000001
(f) 901153
Answer:
(a) 5445
Sum of the digits at odd places = 4 + 5 = 9
Sum of the digits at even places = 4 + 5 = 9
Difference of both sums = 9 – 9 = 0
Since the difference is 0, therefore, the number is divisible by 11.
(b) 10824
Sum of the digits at odd places = 4 + 8 +1 = 13
Sum of the digits at even places = 2 + 0 = 2
Difference of both sums = 13 – 2 = 11
Since the difference is 11, therefore, the number is divisible by 11.
(c) 7138965
Sum of the digits at odd places = 5 + 9 + 3 + 7 = 24
Sum of the digits at even places = 6 + 8 + 1 = 15
Difference of both sums = 24 – 15 = 9
Since the difference is neither 0 nor 11, therefore, the number is not divisible by 11.
(d) 70169308
Sum of the digits at odd places = 8 + 3 + 6 + 0 = 17
Sum of the digits at even places = 0 + 9 + 1 + 7 = 17
Difference of both sums = 17 – 17 = 0
Since the difference is 0, therefore, the number is divisible by 11.
(e) 10000001
Sum of the digits at odd places = 1 + 0 + 0 + 0 = 1
Sum of the digits at even places = 0 + 0 + 0 + 1 = 1
Difference of both sums = 1 – 1 = 0
Since the difference is 0, therefore, the number is divisible by 11.
(f) 901153
Sum of the digits at odd places = 3 + 1 + 0 = 4
Sum of the digits at even places = 5 + 1 + 9 = 15
Difference of both sums = 15 – 4 = 11
Since the difference is 11, therefore, the number is divisible by 11.
Question 5. Write the smallest digit and the largest digit in the blanks space of each of the following numbers, so that the number formed is divisibly by 3:
(a) ____ 6724
(b) 4765 ____ 2
Answer:
(a) We know that a number is divisible by 3 if the sum of all digits is divisible by 3.
Therefore, Smallest digit : 2 26724 = 2 + 6 + 7 + 2 + 4 = 21
Largest digit : 8 86724 = 8 + 6 + 7 + 2 + 4 = 27
(b) We know that a number is divisible by 3 if the sum of all digits is divisible by 3.
Therefore, Smallest digit : 0 476502 = 4 + 7 + 6 + 5 + 0 + 2 = 24
Largest digit : 9 476592 = 4 + 7 + 6 + 5 + 0 + 2 = 33
Question 6. Write the smallest digit and the largest digit in the blanks space of each of the following numbers so that the number formed is divisibly by 11:
(a) 92 ____ 389
(b) 8 ____9484
Answer:
(a) We know that a number is divisible by 11 if the difference of the sum of the digits at odd
places and that of even places should be either 0 or 11.
Therefore, 928389
Odd places = 9 + 8 + 8 = 25
Even places = 2 + 3 + 9 = 14
Difference = 25 – 14 = 11
(b) We know that a number is divisible by 11 if the difference of the sum of the digits at odd
places and that of even places should be either 0 or 11.
Therefore, 869484
Odd places = 8 + 9 + 8 = 25
Even places = 6 + 4 + 4 = 14
Difference = 25 – 14 = 11