Solution 1:
We know numbers having 0, 2, 4, 6 and 8 in ones place are divisible by 2.
(i) The given number = 2650
Digit at unit's place = 0
∴ It is divisible by 2.
(ii) The given number = 69435
Digit at unit's place = 5
∴ It is not divisible by 2.
(iii) The given number = 59628
Digit at unit's place = 8
∴ It is divisible by 2.
(iv) The given number = 789403
Digit at unit's place = 3
∴ It is not divisible by 2.
(v) The given number = 357986
Digit at unit's place = 6
∴ It is divisible by 2.
(vi) The given number = 367314
Digit at unit's place = 4
It is divisible by 2.
Solution 2:
(i) The given number = 733
Sum of its digits = 7 + 3 + 3 = 13,
Since, 13 is not divisible by 3.
∴ 733 is not divisible by 3.
(ii) The given number = 10038
Sum of its digits = 1 + 0 + 0 + 3 + 8 = 12,
Since, 12 is divisible by 3
∴ 10038 is divisible by 3.
(iii) The given number = 20701
Sum of its digits = 2 + 0 + 7 + 0 + 1 = 10,
Since, 10 is not divisible by 3
∴ 20701 is not divisible by 3.
(iv) The given number = 524781
Sum of its digits = 5 + 2 + 4 + 7 + 8 + 1 = 27,
Since, 27 is divisible by 3
∴ 524781 is divisible by 3.
(v) The given number = 79124
Sum of its digits = 7 + 9 + 1 + 2 + 4 = 23,
Since, 23 is not divisible by 3
∴ 79124 is not divisible by 3.
(vi) The given number = 872645
Sum of its digits = 8 + 7 + 2 + 6 + 4 + 5 = 32,
Since, 13 is not divisible by 3
∴ 872645 is not divisible by 3.
Solution 3:
A number is divisible by 4 if the number formed by the digits in its tens and units place is divisible by 4.
(i) The given number = 618
Last 2 digits = 18
18 ÷ 4 = 4.5
That means not divisible.
∴ 618 is not divisible by 4.
(ii) The given number = 2314
Last 2 digits = 14
14 ÷ 4 = 3.5
That means not divisible.
∴ 2314 is not divisible by 4.
(iii) The given number = 63712
Last 2 digits = 12
12 ÷ 4 = 3
That means divisible.
∴ 63712 is divisible by 4.
(iv) The given number = 35056
Last 2 digits = 56
56 ÷ 4 = 14
That means divisible.
∴ 35056 is divisible by 4.
(v) The given number = 946126
Last 2 digits = 26
26 ÷ 4 = 6.5
That means not divisible.
∴ 946126 is not divisible by 4.
(vi) The given number = 810524
Last 2 digits = 24
24 ÷ 4 = 6
That means divisible.
∴ 810524 is divisible by 4.
Solution 4:
We know that a number is divisible by 5 if its ones digit is 0 or 5.
(i) 4965 is divisible by 5, because the digit at its ones place is 5.
(ii) 23590 is divisible by 5, because the digit at its ones place is 0.
(iii) 35208 is not divisible by 5, because the digit at its ones place is 8.
(iv) 723405 is divisible by 5, because the digit at its ones place is 5.
(v) 124684 is not divisible by 5, because the digit at its ones place is 4.
(vi) 438750 is divisible by 5, because the digit at its ones place is 0.
Solution 5:
A number is divisible by 6 if it is divisible by both 2 and 3.
(i) The given number = 2070
Its unit's digit = 0
So, it is divisible by 2
Sum of its digits = 2 + 0 + 7 + 0 = 9,
Since, 9 is divisible by 3
∴ 2070 is divisible by 3.
So, 2070 is divisible by both 2 and 3
Hence it is divisible by 6.
(ii) The given number = 46523
Its unit's digit = 3
So, it is not divisible by 2
Hence 46523 is not divisible by 6.
(iii) The given number = 71232
Its unit's digit = 2
So, it is divisible by 2
Sum of its digits = 7 + 1 + 2 + 3 + 2 = 15, which is divisible by 3
∴ 71232 is divisible by both 2 and 3
Hence it is divisible by 6.
(iv) The given number = 934706
Its unit's digit = 6, so
It is divisible by 2
Sum of its digits = 9 + 3 + 4 + 7 + 0 + 6 = 29, which is not divisible by 3
Hence 934706 is not divisible by 6.
(v) The given number = 251780
Its unit's digit = 0
So, it is divisible by 2
Sum of its digits = 2 + 5 + 1 + 7 + 8 + 0 = 23, which is not divisible by 3
∴ 251780 is not divisible by 6.
(vi) 872536 is not divisible by 6 as sum of its digits is 8 + 7 + 2 + 5 + 3 + 6 = 31 is not divisible by 3
∴ 872536 is not divisible by 6.
Solution 6:
To determine if a number is divisible by 7, double the last digit of the number and subtract it from the number formed by the remaining digits. If their difference is a multiple of 7, the number is divisible by 7.
(i) 826 is divisible by 7.
We have 82 - 2 × 6 = 70, which is divisible by 7.
∴ 826 is divisible by 7.
(ii) 117 is not divisible by 7.
We have 11 - 2 × 7 = -3, which is not divisible by 7
∴ 117 is not divisible by 7.
(iii) 2345 is divisible by 7.
We have 234 - 2 × 5 = 224, which is divisible by 7
∴ 2345 is divisible by 7.
(iv) 6021 is divisible by 7.
We have 602 - 2 × 1 = 600, which is not divisible by 7
∴ 6021 is not divisible by 7.
(v) 14126 is divisible by 7.
We have 1412 - 2 × 6 = 1400, which is divisible by 7
∴ 14126 is divisible by 7.
(vi) 25368 is divisible by 7.
We have 2536 - 2 × 8 = 2520, which is divisible by 7
∴ 25368 is divisible by 7.
Solution 7:
A number is divisible by 8 if the number formed by the last three digits (digits in the hundreds, tens and unit places) is divisible by 8.
(i) 9364
Last 3 digit = 364
364 ÷ 8 = 45.5, it means 364 is not completely divisible by 8.
∴ 9364 is not divisible by 8.
(ii) 20138
Last 3 digits = 138
138 ÷ 8 = 17.25, it means 138 is not completely divisible by 8.
∴ 20138 is not divisible by 8.
(iii) 36792
Last 3 digits = 792
792 ÷ 8 = 99, it means 792 is completely divisible by 8.
∴ 36792 is divisible by 8.
(iv) 901674
Last 3 digits = 674
674 ÷ 8 = 84.25, it means 674 is not completely divisible by 8.
∴ 901674 is not divisible by 8.
(v) 136976
Last 3 digits = 976
976 ÷ 8 = 122, it means 976 is completely divisible by 8.
∴ 136976 is divisible by 8.
(vi) 1790184
Last 3 digits = 184
184 ÷ 8 = 23, it means 184 is completely divisible by 8.
∴ 1790184 is divisible by 8.
Solution 8:
A number is divisible by 9 if the sum of its digits is divisible by 9.
(i) 2358
Sum of its digits = 2 + 3 + 5 + 8 = 18, which is divisible by 9.
∴ 2358 is divisible by 9.
(ii) 3333
Sum of its digits = 3 + 3 + 3 + 3 = 12, which is not divisible by 9.
∴ 3333 is not divisible by 9.
(iii) 98712
Sum of its digits = 9 + 8 + 7 + 1 + 2 = 27, which is divisible by 9.
∴ 98712 is divisible by 9.
(iv) 257106
Sum of its digits = 2 + 5 + 7 + 1 + 0 + 6 = 21, which is not divisible by 9.
∴ 257106 is not divisible by 9.
(v) 647514
Sum of its digits = 6 + 4 + 7 + 5 + 1 + 4 = 27, which is divisible by 9.
∴ 647514 is divisible by 9.
(vi) 326999
Sum of its digits = 3 + 2 + 6 + 9 + 9 + 9 = 38, which is not divisible by 9.
∴ 326999 is not divisible by 9.
Solution 9:
We know that a number is divisible by 10 if its ones digit is 0. Therefore
(i) 48570 is divisible by 10
(ii) 632150 is divisible by 10
(iii) 70005 is not divisible by 10
Solution 10:
A number is divisible by 11 if the difference of the sum of its digits at odd places and the sum of its digits at even places is either 0 or a multiple of 11.
(i) The given number = 403403
Sum of its digits in odd places = 4 + 3 + 0 = 7
Sum of its digits in even places = 0 + 4 + 3 = 7
Difference of the two sums = 7 - 7 = 0
∴ 403403 is divisible by 11.
(ii) The given number = 637219
Sum of its digits in odd places = 6 + 7 + 1 = 14
Sum of its digits in even places = 3 + 2 + 9 = 14
Difference of the two sums = 14 - 14 = 0
∴ 637219 is divisible by 11.
(iii) The given number = 975488
Sum of its digits in odd places = 9 + 5 + 8 = 22
Sum of its digits in even places = 7 + 4 + 8 = 19
Difference of the two sums = 22 - 19 = 3
∴ 975488 is not divisible by 11.
(iv) The given number = 137269
Sum of its digits in odd places = 1 + 7 + 6 = 14
Sum of its digits in even places = 3 + 2 + 9 = 14
Difference of the two sums = 14 - 14 = 0
∴ 137269 is divisible by 11.
(v) The given number = 901351
Sum of its digits in odd places = 9 + 1 + 5 = 15
Sum of its digits in even places = 0 + 3 + 1 = 4
Difference of the two sums = 15 - 4 = 11
∴ 901351 is divisible by 11.
(vi) The given number = 8790322
Sum of its digits in odd places = 8 + 9 + 3 + 2 = 22
Sum of its digits in even places = 7 + 0 + 2 = 9
Difference of the two sums = 22 - 9 = 13
∴ 8790322 is not divisible by 11.
(vii) The given number = 5683249
Sum of its digits in odd places = 5 + 8 + 2 + 9 = 24
Sum of its digits in even places = 6 + 3 + 4 = 13
Difference of the two sums = 24 - 13 = 11
∴ 5683249 is divisible by 11.
(viii) The given number = 3576958
Sum of its digits in odd places = 3 + 7 + 9 + 8 = 27
Sum of its digits in even places = 5 + 6 + 5 = 16
Difference of the two sums = 27 - 16 = 11
∴ 3576958 is divisible by 11.
(ix) The given number = 479866
Sum of its digits in odd places = 4 + 9 + 6 = 19
Sum of its digits in even places = 7 + 8 + 6 = 21
Difference of the two sums = 19 - 21 = -2
∴ 479866 is not divisible by 11.
Solution 11:
(i) The given number = 57 × 4
Sum of known digits = 5 + 7 + 4 = 16
Next multiple of 3 after 16 is 18
∴ Required smallest number = 18 - 16 = 2
(ii) The given number = 53 × 46
Sum of known digits = 5 + 3 + 4 + 6 = 18
Now, 18 is divisible by 3.
∴ Required smallest number = 0
(iii) The given number = 8 × 725
Sum of known digits = 8 + 7 + 2 + 5 = 22
Next multiple of 3 after 22 is 24
∴ Required smallest number = 24 - 22 = 2
(iv) The given number = 62 × 357
Sum of known digits = 6 + 2 + 3 + 5 + 7 = 23
Next multiple of 3 after 23 is 24
∴ Required smallest number = 24 - 23 = 1
(v) The given number = 234 × 178
Sum of known digits = 2 + 3 + 4 + 1 + 7 + 8 = 25
Next multiple of 3 after 25 is 27
∴ Required smallest number = 27 - 25 = 2
(vi) The given number = 6 × 10543
Sum of known digits = 6 + 1 + 0 + 5 + 4 + 3 = 19
Next multiple of 3 after 19 is 21
∴ Required smallest number = 21 - 19 = 2
Solution 12:
(i) The given number = 65 × 5
Sum of its given digits = 6 + 5 + 5 = 16
The number next to 17, which is divisible by 9 is 18.
∴ Required smallest number = 18 - 16 = 2
(ii) The given number = 2 × 135
Sum of its given digits = 2 + 1 + 3 + 5 = 11
The number next to 11, which is divisible by 9 is 18.
∴ Required smallest number = 18 - 11 = 7
(iii) The given number = 6702 ×
Sum of its given digits = 6 + 7 + 0 + 2 = 15
The number next to 15, which is divisible by 9 is 18.
∴ Required smallest number = 18 - 15 = 3
(iv) The given number = 91 × 67
Sum of its given digits = 9 + 1 + 6 + 7 = 23
The number next to 23, which is divisible by 9 is 27.
∴ Required smallest number = 27 - 23 = 4
(v) The given number = 6678 × 1
Sum of its given digits = 6 + 6 + 7 + 8 + 1 = 28
The number next to 28, which is divisible by 9 is 36.
∴ Required smallest number = 36 - 28 = 8
(vi) The given number = 835 × 86
Sum of its given digits = 8 + 3 + 5 + 8 + 6 = 30
The number next to 30, which is divisible by 9 is 36.
∴ Required smallest number = 36 - 30 = 6
Solution 13:
(i) The given number = 26 × 5
Sum of its digits in odd places = 5 + 6 = 11
Sum of its digits in even places = * + 2
Difference = Sum of odd digits - Sum of digits in even places
= 11 - (* + 2)
The given number will be divisible by 11 if the difference of the two sums = 0.
∴ 11 - (* + 2) = 0
⇒ 11 = * + 2
⇒ * = 11 - 2
⇒ * = 9
∴ Required smallest number = 9
(ii) The given number = 39 × 43
Sum of its digits in odd places = 3 + * + 3 = * + 6
Sum of its digits in even places = 4 + 9 = 13
Difference = Sum of odd digits - Sum of digits in even places
= * + 6 - 13
= * - 7
The given number will be divisible by 11, if the difference of the two sums = 0.
∴ * - 7 = 0
⇒ * = 7
Hence, required smallest number = 7.
(iii) The given number = 86 × 72
Sum of its digits in odd places = 2 + * + 8 = * + 10
Sum of its digits in even places = 7 + 6 = 13
Difference = Sum of odd digits - Sum of digits in even places
= * + 10 - 13
= * - 3
The given number will be divisible by 11, if the difference of the two sums = 0.
∴ * - 3 = 0
⇒ * = 3
Hence, required smallest number = 3.
(iv) The given number = 467 × 91
Sum of its digits in odd places = 1 + * + 6 = * + 7
Sum of its digits in even places = 9 + 7 + 4 = 20
Difference = Sum of odd digits - Sum of digits in even places
= 20 - (* + 7)
= 20 - * - 7
= 13 - *
Clearly the difference of the two sums will be multiple of 11 if
13 - * = 11
⇒ 13 - 11 = *
⇒ 2 = *
⇒ * = 2
Hence, required smallest number = 2.
(v) The given number = 1723 × 4
Sum of its digits in odd places = 4 + 3 + 7 = 14
Sum of its digits in even places = * + 2 + 1 = * + 3
Difference = Sum of odd digits - Sum of digits in even places
= * + 3 - 14 = * - 11
The given number will be divisible by 11, if * - 11 is a multiple of 11, which is possible if * = 0.
∴ Required smallest number = 0.
(vi) The given number = 9 × 8071
Sum of its digits in odd places = 1 + 0 + * = 1 + *
Sum of its digits in even places = 7 + 8 + 9 = 24
Difference = Sum of odd digits - Sum of digits in even places
= 24 - 1 - *
= 23 - *
Now, (* - 23) will be divisible by 11 if * = 1.
i.e., 1 - 23 = -22
-22 is divisible by 11.
∴ * = 1
Hence, required smallest number = 1.
Solution 14:
87930 is divisible by 45 if it is divisible by both 5 and 9
Check divisibility by 5:
The unit digit of 87930 is 0
∴ 87930 is divisible by 5
Check divisibility by 9:
Sum of digits = 8 + 7 + 9 + 3 + 0 = 27, which is divisible by 9
∴ 87930 is divisible by 9.
Hence, 87930 is divisible by both 5 and 9, so 87930 is divisible by 45 also.
Solution 15:
A number between 100 and 200 is a prime number if it is not divisible by any prime number less than 15.
Similarly, a number between 200 and 300 is a prime number if it is not divisible by any prime number less than 20.
(i) 103 is a prime number, because it is not divisible by 2, 3, 5, 7, 11 and 13.
(ii) 137 is a prime number, because it is not divisible by 2, 3, 5, 7 and 11.
(iii) 161 is a not prime number, because it is divisible by 7.
(iv) 179 is a prime number, because it is not divisible by 2, 3, 5, 7, 11 and 13.
(v) 217 is a not prime number, because it is divisible by 7.
(vi) 277 is a prime number, because it is not divisible by 2, 3, 5, 7, 11, 13, 17 and 19.
(vii) 331 is a prime number, because it is not divisible by 2, 3, 5, 7, 11, 13, 17 and 19.
(viii) 397 is a prime number, because it is not divisible by 2, 3, 5, 7, 11, 13, 17 and 19.
Solution 16:
(i) 44
All odd prime numbers less than 44 are
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41 and 43.
Clearly, 44 = (13 + 31) = sum of two odd primes.
(ii) 36
All odd prime numbers less than 36 are
3, 5, 7, 11, 13, 17, 19, 23, 29 and 31
Clearly, 36 = (5 + 31) = sum of two odd primes
(iii) 24
All odd prime numbers less than 24 are
3, 5, 7, 11, 13, 17, 19 and 23
Clearly, 24 = (5 + 19) = sum of two odd primes
(iv) 18
All odd prime numbers less than 18 are
3, 5, 7, 11, 13 and 17
Clearly, 18 = (5 + 13) = sum of two odd primes
Solution 17:
The four smallest prime numbers are 2, 3, 5, 7
Multiply them: 2 × 3 × 5 × 7 = 210
∴ 210 is the smallest number that has four different prime factors.
Solution 18:
(i) 35
(ii) 21
(iii) 24
(iv) 30
Solution 19:
(i) False
Reason: 12 is divisible by 4 but not by 8.
(ii) True
(iii) False
Reason: 5 divides (2 + 3) = 5, but does not divide 2 or 3 individually.
(iv) True
(v) True
(vi) True
(vii) True
