Unit-1: Factors and Multiples
EXERCISE 1.2
1. State which of the following are divisible by 2, 3, 4 or 5 and give a reason
Solutions:
26 is divisible by 2 as its ending digit is 6.
26 is not divisible by 3 as the sum of its all digits i.e 2+6 = 8 is not divisible by 3.
26 is not divisible by 4 as the two digits i.e 26 is not divisible by 4.
26 is not divisible by 5 as its ending digit is not 0 or 5.
Thus
26 is divisible by 2 -----------Ans. (i)
63 is not divisible by 2 as its ending digit is not 0, 2, 4, 6 or 8.
63 is divisible by 3 as the sum of its all digits i.e 6+3 = 9 is divisible by 3.
63 is not divisible by 4 as the two digits i.e 63 is not divisible by 4.
63 is not divisible by 5 as its ending digit is not 0 or 5.
Thus
63 is divisible by 3 ----------Ans. (ii)
110 is divisible by 2 as its ending digit is 0.
110 is not divisible by 3 as the sum of its all digits i.e 1+1+0 = 2 is not divisible by 3.
110 is not divisible by 4 as the last two digits i.e 10 is not divisible by 4.
110 is divisible by 5 as its ending digit is 0.
Thus
110 is divisible by 2 and 5 -----------Ans. (iii)
435 is not divisible by 2 as its ending digit is not 0, 2, 4, 6 or 8.
435 is divisible by 3 as the sum of its all digits i.e 4+3+5 = 12 is divisible by 3.
435 is not divisible by 4 as the last two digits i.e 35 is not divisible by 4.
435 is divisible by 5 as its ending digit is 5.
Thus
435 is divisible by 3 and 5 -----------Ans. (iv)
6123 is not divisible by 2 as its ending digit is not 0, 2, 4, 6 or 8.
6123 is divisible by 3 as the sum of its all digits i.e 6+1+2+3 = 12 is divisible by 3.
6123 is not divisible by 4 as the last two digits i.e 23 is not divisible by 4.
6123 is not divisible by 5 as its ending digit is not 0 or 5.
Thus
1036 is divisible by 2 as its ending digit is 6.
1036 is not divisible by 3 as the sum of its all digits i.e 1+0+3+6 = 10 is not divisible by 3.
1036 is divisible by 4 as the last two digits i.e 36 is divisible by 4.
1036 is not divisible by 5 as its ending digit is not 0 or 5.
Thus
3000 is divisible by 2 as its ending digit is 0.
3000 is divisible by 3 as the sum of its all digits i.e 3+0+0+0 = 3 is divisible by 3.
3000 is divisible by 4 as the last two digits i.e 00 is divisible by 4.
3000 is divisible by 5 as its ending digit is 0 .
Thus
2. State which of the following are divisible by 6, 8, 9,10 or 11 and give a reason
(i) 7170 (ii) 4128 (iii) 9900(iv) 43212 (v) 4455 (vi) 7172(vii) 619291 (iv) 24120
Solutions:
(i) 7170
7170 is divisible by 6 as it is divisible by both 2 and 3.
7170 is NOT divisible by 8 as the last three digits i.e 170 is not divisible by 8.
7170 is NOT divisible by 9 as the sum of its all digits i.e 7+1+7+0 = 15 is not divisible by 9.
7170 is divisible by 10 as its ending digit is 0.
7170 is NOT divisible by 11 as (7+7) - (1+0) = 14 - 1 = 13 ≠ 0 or multiple of 11.
Thus
7170 is divisible by 6 and 10 -----------Ans. (i)
(ii) 4128
4128 is divisible by 6 as it is divisible by both 2 and 3.
4128 is divisible by 8 as the last three digits i.e 128 is divisible by 8.
4128 is NOT divisible by 9 as the sum of its all digits i.e 4+1+2+8 = 15 is not divisible by 9.
4128 is NOT divisible by 10 as its ending digit is not a 0.
4128 is NOT divisible by 11 as (4+2) - (1+8) = 6 - 9 = -3 ≠ 0 or multiple of 11.
Thus
4128 is divisible by 6 and 8 -----------Ans. (ii)
(iii) 9900
9900 is divisible by 6 as it is divisible by both 2 and 3.
9900 is NOT divisible by 8 as the last three digits i.e 990 is not divisible by 8.
9900 is divisible by 9 as the sum of its all digits i.e 9+9+0+0 = 18 is divisible by 9.
9900 is divisible by 10 as its ending digit is 0.
9900 is divisible by 11 as (9+0) - (9+0) = 9 - 9 = 0
Thus
9900 is divisible by 6 9, 10 and 11 -----------Ans. (iii)
(iv) 43212
43212 is divisible by 6 as it is divisible by both 2 and 3.
43212 is NOT divisible by 8 as the last three digits i.e 212 is not divisible by 8.
43212 is NOT divisible by 9 as the sum of its all digits i.e 4+3+2+1+2 = 12 is not divisible by 9.
43212 is NOT divisible by 10 as its ending digit is not a 0.
43212 is NOT divisible by 11 as (4+2+2) - (3+2) = 8 - 5 = 3 ≠ 0 or multiple of 11.
Thus
43212 is divisible by 6 -----------Ans. (iii)
(v) 4455
4455 is NOT divisible by 6 as it is divisible by 2 and but not by 3.
4455 is NOT divisible by 8 as the last three digits i.e 455 is not divisible by 8.
4455 is divisible by 9 as the sum of its all digits i.e 4+4+5+5 = 18 is divisible by 9.
4455 is NOT divisible by 10 as its ending digit is not a 0.
4455 is divisible by 11 as (4+5) - (4+5) = 9 - 9 = 0
Thus
4455 is divisible by 9 and 11 -----------Ans. (iv)
(vi) 7172
7172 is NOT divisible by 6 as it is divisible by 2 and but not by 3.
7172 is NOT divisible by 8 as the last three digits i.e 172 is not divisible by 8.
7172 is NOT divisible by 9 as the sum of its all digits i.e 7+1+7+2 = 17 is not divisible by 9.
7172 is NOT divisible by 10 as its ending digit is not a 0.
7172 is divisible by 11 as (7+7) - (1+2) = 14 - 3 = 11 (multiple of 11).
Thus
7172 is divisible by 11 -----------Ans. (vi)
(vii) 619291
619291 is NOT divisible by 6 as it is not divisible by 2 and 3.
619291 is NOT divisible by 8 as the last three digits i.e 291 is not divisible by 8.
619291 is NOT divisible by 9 as the sum of its all digits i.e 6+1+9+2+9+2 = 29 is not divisible by 9.
619291 is NOT divisible by 10 as its ending digit is not a 0.
619291 is NOT divisible by 11 as (6+9+9) - (1+2+1) = 24 - 4 = 20 ≠ 0 or multiple of 11.
Thus
619291 is NOT divisible by 6,8,9,10 or 11 -----------Ans. (vii)
(viii) 24120
24120 is divisible by 6 as it is divisible by both 2 and 3.
24120 is divisible by 8 as the last three digits i.e 120 is divisible by 8.
24120 is divisible by 9 as the sum of its all digits i.e 2+4+1+2+0 = 9 is divisible by 9.
24120 is divisible by 10 as its ending digit is 0.
24120 is NOT divisible by 11 as (2+1+0) - (4+2) = 3 - 4 = -1 ≠ 0 or multiple of 11.
Thus
24120 is divisible by 6 8, 9 and 10 -----------Ans. (viii)
7170 is divisible by 6 as it is divisible by both 2 and 3.
7170 is NOT divisible by 8 as the last three digits i.e 170 is not divisible by 8.
7170 is NOT divisible by 9 as the sum of its all digits i.e 7+1+7+0 = 15 is not divisible by 9.
7170 is divisible by 10 as its ending digit is 0.
7170 is NOT divisible by 11 as (7+7) - (1+0) = 14 - 1 = 13 ≠ 0 or multiple of 11.
Thus
7170 is divisible by 6 and 10 -----------Ans. (i)
4128 is divisible by 6 as it is divisible by both 2 and 3.
4128 is divisible by 8 as the last three digits i.e 128 is divisible by 8.
4128 is NOT divisible by 9 as the sum of its all digits i.e 4+1+2+8 = 15 is not divisible by 9.
4128 is NOT divisible by 10 as its ending digit is not a 0.
4128 is NOT divisible by 11 as (4+2) - (1+8) = 6 - 9 = -3 ≠ 0 or multiple of 11.
Thus
9900 is divisible by 6 as it is divisible by both 2 and 3.
9900 is NOT divisible by 8 as the last three digits i.e 990 is not divisible by 8.
9900 is divisible by 9 as the sum of its all digits i.e 9+9+0+0 = 18 is divisible by 9.
9900 is divisible by 10 as its ending digit is 0.
9900 is divisible by 11 as (9+0) - (9+0) = 9 - 9 = 0
Thus
43212 is divisible by 6 as it is divisible by both 2 and 3.
43212 is NOT divisible by 8 as the last three digits i.e 212 is not divisible by 8.
43212 is NOT divisible by 9 as the sum of its all digits i.e 4+3+2+1+2 = 12 is not divisible by 9.
43212 is NOT divisible by 10 as its ending digit is not a 0.
43212 is NOT divisible by 11 as (4+2+2) - (3+2) = 8 - 5 = 3 ≠ 0 or multiple of 11.
Thus
4455 is NOT divisible by 6 as it is divisible by 2 and but not by 3.
4455 is NOT divisible by 8 as the last three digits i.e 455 is not divisible by 8.
4455 is divisible by 9 as the sum of its all digits i.e 4+4+5+5 = 18 is divisible by 9.
4455 is NOT divisible by 10 as its ending digit is not a 0.
4455 is divisible by 11 as (4+5) - (4+5) = 9 - 9 = 0
Thus
7172 is NOT divisible by 6 as it is divisible by 2 and but not by 3.
7172 is NOT divisible by 8 as the last three digits i.e 172 is not divisible by 8.
7172 is NOT divisible by 9 as the sum of its all digits i.e 7+1+7+2 = 17 is not divisible by 9.
7172 is NOT divisible by 10 as its ending digit is not a 0.
7172 is divisible by 11 as (7+7) - (1+2) = 14 - 3 = 11 (multiple of 11).
Thus
619291 is NOT divisible by 6 as it is not divisible by 2 and 3.
619291 is NOT divisible by 8 as the last three digits i.e 291 is not divisible by 8.
619291 is NOT divisible by 9 as the sum of its all digits i.e 6+1+9+2+9+2 = 29 is not divisible by 9.
619291 is NOT divisible by 10 as its ending digit is not a 0.
619291 is NOT divisible by 11 as (6+9+9) - (1+2+1) = 24 - 4 = 20 ≠ 0 or multiple of 11.
Thus
24120 is divisible by 6 as it is divisible by both 2 and 3.
24120 is divisible by 8 as the last three digits i.e 120 is divisible by 8.
24120 is divisible by 9 as the sum of its all digits i.e 2+4+1+2+0 = 9 is divisible by 9.
24120 is divisible by 10 as its ending digit is 0.
24120 is NOT divisible by 11 as (2+1+0) - (4+2) = 3 - 4 = -1 ≠ 0 or multiple of 11.
Thus
3. From the numbers 5, 8, 81, 85, 60, 26, 54, 45. Write numbers divisible by 5
Solutions:
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4. Is 6 a factor of 432 ? Give reason.
Hint: Factor of a number are the numbers that it can be divided exactly by:
Solution:
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