Chapter 3 Playing With Numbers Ex 3.3 Solutions
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 |
| | | | | | | | | |
| 128 990 1586 275 6686 639210 429714 2856 3060 406839 | Yes | No | Yes | No | No | Yes | No | No | No |
Answer - 1 : -
| 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) 5500, (d) 6000, (e) 12159, (f) 14560, (g) 21084, (h) 31795072, (i) 1700, (j) 2150
Answer - 2 : -
(a) 572 → 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 → Not divisible by 4 as its last two digits are not divisible by 4.
→ 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, (ij) 17852
Answer - 3 : -
(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 an even number.
→ Divisible by 3 as sum of its digits (= 27) is not 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 not 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 an even number.
→ Divisible by 3 as sum of its digits (= 21) is not divisible by 3.
Since the number is divisible by both 2 and 3, therefore, it is divisible by 6.
(ji) 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 - 4 : -
(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 - 5 : -
(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 - 6 : -
(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