Fill in the blanks:
(i) The sum of a 2-digit number and number obtained by reversing the digits is always divisible by 11 and .....
(ii) The difference of a 2-digit number and number obtained by reversing the digits is always divisible by ..... and .....
(iii) The difference of a 3-digit number abc (a > c) and number obtained by reversing the digits is always divisible by ..... and .....
(iv) The next number of the series 0, 1, 1, 2, 3, 5, 8, 13, ..... is .....
(v) General form of a 2 digit number 57 is .....
(vi) Usual form of number 100 × 7 + 10 × 4 + 1 is .....
(vii) If the division N ÷ 5 leaves remainder 4 and the division N ÷ 2 leaves remainder 1, then unit's digit of N must be .....
(viii) If 213x52 is divisible by 9, then digit x is .....
(ix) The number ab - ba where a, b are digits and a > b is divisible by .....
Answer
(i) The sum of a 2-digit number and number obtained by reversing the digits is always divisible by 11 and sum of the digits.
(ii) The difference of a 2-digit number and number obtained by reversing the digits is always divisible by 9 and difference of the digits.
(iii) The difference of a 3-digit number abc (a > c) and number obtained by reversing the digits is always divisible by 99 and (a - c).
(iv) The next number of the series 0, 1, 1, 2, 3, 5, 8, 13, ..... is 21.
(v) General form of a 2 digit number 57 is 10 × 5 + 7.
(vi) Usual form of number 100 × 7 + 10 × 4 + 1 is 741.
(vii) If the division N ÷ 5 leaves remainder 4 and the division N ÷ 2 leaves remainder 1, then unit's digit of N must be 9.
(viii) If 213x52 is divisible by 9, then digit x is 5.
(ix) The number ab - ba where a, b are digits and a > b is divisible by 9.
State whether the following statements are true (T) or false (F):
(i) If the division N ÷ 5 leaves a remainder 4, then unit's digit of N may be 1 or 6.
(ii) If the division N ÷ 2 leaves a remainder 1, then unit's digit of N may be 1, 3, 5, 7 or 9.
(iii) If a number is divisible by any number m, then it will also be divisible by each of the factors of m.
(iv) If a number is divisible by sum of two numbers then it will also be divisible by each of the two numbers separately.
(v) If 3651x is divisible by 9, then value of digit x is 3
(vi) If 42x7 is divisible by 3, then value of digit x is 4
(vii) The number 152875 is divisible by 9
(viii) The number 389072 is divisible by 8
(ix) The number 27402 is divisible by 6
(x) The number 359249 is not divisible by 11
Answer
(i) False
If the division N ÷ 5 leaves a remainder 4, then the unit's digit of N may be 4 or 9.
(ii) True
If the division N ÷ 2 leaves a remainder 1, then N is an odd number, so its unit's digit may be 1, 3, 5, 7 or 9.
(iii) True
If a number is divisible by any number m, then it will also be divisible by each of the factors of m.
(iv) False
For example, 5 is divisible by the sum 2 + 3 = 5, but 5 is neither divisible by 2 nor by 3.
(v) True
Sum of digits of 3651x = 3 + 6 + 5 + 1 + x = 15 + x, which is divisible by 9 when 15 + x = 18 i.e. x = 3.
(vi) False
Sum of digits of 42x7 = 4 + 2 + x + 7 = 13 + x, which is divisible by 3 when 13 + x = 15, 18, 21 i.e. x = 2, 5 or 8. So, x cannot be 4.
(vii) False
Sum of digits of 152875 = 1 + 5 + 2 + 8 + 7 + 5 = 28, which is not divisible by 9.
(viii) True
Number formed by last three digits of 389072 = 072 = 8 × 9, which is divisible by 8.
(ix) True
Last digit of 27402 is 2, so it is divisible by 2. Sum of digits = 2 + 7 + 4 + 0 + 2 = 15, which is divisible by 3, so it is divisible by 3. Hence, 27402 is divisible by 6.
(x) False
For the number 359249, sum of digits at odd places = 9 + 2 + 5 = 16 and sum of digits at even places = 4 + 9 + 3 = 16. Their difference = 16 - 16 = 0. Hence, 359249 is divisible by 11.