Which of the following numbers are divisible by 5 or by 10:
(i) 87035
(ii) 75060
(iii) 9685
(iv) 10730
Answer
According to the divisibility of 5, if the unit's digit of the number is 0 or 5 then the number is divisible by 5.
According to the divisibility of 10, if the unit's digit of the number is 0 then the number is divisible by 10.
(i) 87035 — Last digit is 5. So, it is divisible by 5 but not by 10.
(ii) 75060 — Last digit is 0. So, it is divisible by both 5 and 10.
(iii) 9685 — Last digit is 5. So, it is divisible by 5 but not by 10.
(iv) 10730 — Last digit is 0. So, it is divisible by both 5 and 10.
Hence, the numbers (i), (ii), (iii) and (iv) are divisible by 5 and the numbers (ii) and (iv) are divisible by 10.
Which of the following numbers are divisible by 2, 4 or 8:
(i) 67894
(ii) 5673244
(iii) 9685048
(iv) 6533142
(v) 75379
Answer
According to the divisibility of 2, if the unit's digit of the number is 0, 2, 4, 6 or 8 then the number is divisible by 2.
According to the divisibility of 4, if the number formed by its last two (ten's and unit's) digits is divisible by 4 then the number is divisible by 4.
According to the divisibility of 8, if the number formed by its last three (hundred's, ten's and unit's) digits is divisible by 8 then the number is divisible by 8.
(i) 67894 — Last digit is 4, so it is divisible by 2. Number formed by last two digits = 94, which is not divisible by 4. So, it is neither divisible by 4 nor by 8.
(ii) 5673244 — Last digit is 4, so it is divisible by 2. Number formed by last two digits = 44 = 4 × 11, so it is divisible by 4. Number formed by last three digits = 244, which is not divisible by 8. So, it is not divisible by 8.
(iii) 9685048 — Last digit is 8, so it is divisible by 2. Number formed by last two digits = 48 = 4 × 12, so it is divisible by 4. Number formed by last three digits = 048 = 8 × 6, so it is divisible by 8.
(iv) 6533142 — Last digit is 2, so it is divisible by 2. Number formed by last two digits = 42, which is not divisible by 4. So, it is neither divisible by 4 nor by 8.
(v) 75379 — Last digit is 9, which is an odd number. So, it is not divisible by 2, 4 or 8.
Hence, the numbers (i), (ii), (iii) and (iv) are divisible by 2, the numbers (ii) and (iii) are divisible by 4 and the number (iii) is divisible by 8.
Which of the following numbers are divisible by 3 or 9:
(i) 45639
(ii) 301248
(iii) 567081
(iv) 345903
(v) 345046
Answer
According to the divisibility of 3, if the sum of the digits of the number is divisible by 3 then the number is divisible by 3.
According to the divisibility of 9, if the sum of the digits of the number is divisible by 9 then the number is divisible by 9.
(i) 45639 — Sum of digits = 4 + 5 + 6 + 3 + 9 = 27, which is divisible by both 3 and 9.
(ii) 301248 — Sum of digits = 3 + 0 + 1 + 2 + 4 + 8 = 18, which is divisible by both 3 and 9.
(iii) 567081 — Sum of digits = 5 + 6 + 7 + 0 + 8 + 1 = 27, which is divisible by both 3 and 9.
(iv) 345903 — Sum of digits = 3 + 4 + 5 + 9 + 0 + 3 = 24, which is divisible by 3 but not by 9.
(v) 345046 — Sum of digits = 3 + 4 + 5 + 0 + 4 + 6 = 22, which is neither divisible by 3 nor by 9.
Hence, the numbers (i), (ii), (iii) and (iv) are divisible by 3 and the numbers (i), (ii) and (iii) are divisible by 9.
Which of the following numbers are divisible by 11:
(i) 10835
(ii) 380237
(iii) 504670
(iv) 28248
Answer
According to the divisibility of 11, if the difference of the sum of digits at the odd places (starting from unit's place) and the sum of the digits at even places is either 0 or divisible by 11, then the number is divisible by 11.
(i) 10835
Sum of digits at odd places = 5 + 8 + 1 = 14
Sum of digits at even places = 3 + 0 = 3
Their difference = 14 - 3 = 11, which is divisible by 11.
∴ 10835 is divisible by 11.
(ii) 380237
Sum of digits at odd places = 7 + 2 + 8 = 17
Sum of digits at even places = 3 + 0 + 3 = 6
Their difference = 17 - 6 = 11, which is divisible by 11.
∴ 380237 is divisible by 11.
(iii) 504670
Sum of digits at odd places = 0 + 6 + 0 = 6
Sum of digits at even places = 7 + 4 + 5 = 16
Their difference = 16 - 6 = 10, which is neither 0 nor divisible by 11.
∴ 504670 is not divisible by 11.
(iv) 28248
Sum of digits at odd places = 8 + 2 + 2 = 12
Sum of digits at even places = 4 + 8 = 12
Their difference = 12 - 12 = 0
∴ 28248 is divisible by 11.
Hence, the numbers (i), (ii) and (iv) are divisible by 11.
Which of the following numbers are divisible by 6:
(i) 15414
(ii) 213888
(iii) 469876
Answer
According to the divisibility of 6, if the number is divisible by both 2 and 3 then the number is divisible by 6.
(i) 15414
Last digit is 4, so it is divisible by 2.
Sum of digits = 1 + 5 + 4 + 1 + 4 = 15, which is divisible by 3, so it is divisible by 3.
∴ 15414 is divisible by 6.
(ii) 213888
Last digit is 8, so it is divisible by 2.
Sum of digits = 2 + 1 + 3 + 8 + 8 + 8 = 30, which is divisible by 3, so it is divisible by 3.
∴ 213888 is divisible by 6.
(iii) 469876
Last digit is 6, so it is divisible by 2.
Sum of digits = 4 + 6 + 9 + 8 + 7 + 6 = 40, which is not divisible by 3, so it is not divisible by 3.
∴ 469876 is not divisible by 6.
Hence, the numbers (i) and (ii) are divisible by 6.
(i) If 34x is a multiple of 3, where x is a digit, what is the value of x?
(ii) If 74x5284 is a multiple of 3, where x is a digit, find the value(s) of x.
Answer
According to the divisibility of 3, if the sum of the digits of the number is divisible by 3 then the number is divisible by 3.
(i) ∵ 34x is a multiple of 3
⇒ 3 + 4 + x is divisible by 3
⇒ 7 + x is divisible by 3
⇒ 7 + x = 0, 3, 6, 9, 12, 15, 18, ...............
⇒ x = -7, -4, -1, 2, 5, 8, 11, ...............
Since x is a digit.
Hence, x = 2, 5 or 8.
(ii) ∵ 74x5284 is a multiple of 3
⇒ 7 + 4 + x + 5 + 2 + 8 + 4 is divisible by 3
⇒ 30 + x is divisible by 3
⇒ 30 + x = 30, 33, 36, 39, 42, ...............
⇒ x = 0, 3, 6, 9, 12, ...............
Since x is a digit.
Hence, x = 0, 3, 6 or 9.
If 42z3 is a multiple of 9, where z is a digit, what is the value of z?
Answer
According to the divisibility of 9, if the sum of the digits of the number is divisible by 9 then the number is divisible by 9.
∵ 42z3 is a multiple of 9
⇒ 4 + 2 + z + 3 is divisible by 9
⇒ 9 + z is divisible by 9
⇒ 9 + z = 9, 18, 27, ...............
⇒ z = 0, 9, 18, ...............
Since z is a digit. So, z = 0 or 9.
In each of the following replace * by a digit so that the number formed is divisible by 9:
(i) 49 * 2207
(ii) 5938 * 623
Answer
According to the divisibility of 9, if the sum of the digits of the number is divisible by 9 then the number is divisible by 9.
(i) ∵ 49*2207 is divisible by 9
⇒ 4 + 9 + * + 2 + 2 + 0 + 7 is divisible by 9
⇒ 24 + * is divisible by 9
⇒ 24 + * = 27, 36, ...............
⇒ * = 3, 12, ...............
Since * is a digit,
∴ * is to be replaced by 3.
The number formed is 4932207, which is divisible by 9.
Hence, * is to be replaced by 3.
(ii) ∵ 5938*623 is divisible by 9
⇒ 5 + 9 + 3 + 8 + * + 6 + 2 + 3 is divisible by 9
⇒ 36 + * is divisible by 9
⇒ 36 + * = 36, 45, 54, ...............
⇒ * = 0, 9, 18, ...............
Since * is a digit, ∴ * is to be replaced by 0 or 9.
The numbers formed are 59380623 and 59389623, which are divisible by 9.
Hence, * is to be replaced by 0 or 9.
In each of the following replace * by a digit so that the number formed is divisible by 6:
(i) 97 * 542
(ii) 709 * 94
Answer
According to the divisibility of 6, if the number is divisible by both 2 and 3 then the number is divisible by 6.
(i) Since the unit's digit of 97*542 is 2, so the number formed is divisible by 2.
Therefore, the number formed will be divisible by 6 if it is divisible by 3 i.e. the sum of its digits is divisible by 3.
⇒ 9 + 7 + * + 5 + 4 + 2 is divisible by 3
⇒ 27 + * is divisible by 3
⇒ 27 + * = 27, 30, 33, 36, 39, ...............
⇒ * = 0, 3, 6, 9, 12, ...............
Since * is a digit.
Hence, * is to be replaced by 0, 3, 6 or 9.
(ii) Since the unit's digit of 709*94 is 4, so the number formed is divisible by 2.
Therefore, the number formed will be divisible by 6 if it is divisible by 3 i.e. the sum of its digits is divisible by 3.
⇒ 7 + 0 + 9 + * + 9 + 4 is divisible by 3
⇒ 29 + * is divisible by 3
⇒ 29 + * = 30, 33, 36, 39, ...............
⇒ * = 1, 4, 7, 10, ...............
Since * is a digit. Hence, * is to be replaced by 1, 4 or 7.
In each of the following replace * by a digit so that the number formed is divisible by 11:
(i) 64 * 2456
(ii) 86 * 6194
Answer
According to the divisibility of 11, if the difference of the sum of digits at the odd places (starting from unit's place) and the sum of the digits at even places is either 0 or divisible by 11, then the number is divisible by 11.
(i) For the number 64*2456,
Sum of digits at odd places = 6 + 4 + * + 6 = 16 + *
Sum of digits at even places = 5 + 2 + 4 = 11
Their difference = 16 + * - 11 = 5 + *, which should be either 0 or divisible by 11.
⇒ 5 + * = 11, 22, ...............
⇒ * = 6, 17, ...............
Since * is a digit.
Hence, * is to be replaced by 6.
(ii) For the number 86*6194,
Sum of digits at odd places = 4 + 1 + * + 8 = 13 + *
Sum of digits at even places = 9 + 6 + 6 = 21
Their difference = 21 - (13 + *) = 8 - *, which should be either 0 or divisible by 11.
⇒ 8 - * = 0, 11, 22, ...............
⇒ * = 8, -3, -14, ...............
Since * is a digit.
Hence, * is to be replaced by 8.