Divisibility Rules(LEVEL - 1)
Q1–10 of 50What least value must be given to n so that the number 6135n2 becomes divisible by 9?
Least value of n so that 6135n2 is divisible by 9 Rule:
A number is divisible by 9 if the sum of its digits
is divisible by 9.
6+1+3+5+n+2=17+n
The smallest value of nn making this a multiple of 9 is:
17+1=18
✅ Answer: (a) 1
Find the multiple of 11 in the following numbers.
Rule: A number is divisible by 11 if the difference between the sum of digits at odd places and even places is 0 or a multiple of 11.
Test options:
(a) $112144 \rightarrow (4+1+1) - (4+2+1) = 6 - 7 = -1$ (Not divisible)
(b) $447355 \rightarrow (5+3+4) - (5+7+4) = 12 - 16 = -4$ (Not divisible)
(c) $869756 \rightarrow (6+7+6) - (5+9+8) = 19 - 22 = -3$ (Not divisible)
(d) $978626 \rightarrow (6+6+7) - (2+8+9) = 19 - 19 = 0$ (Divisible)
Correct Option: (d) 978626
111,111,111,111 is divisible by
$111,111,111,111$ has 12 ones.
Sum of digits $= 12$, which is divisible by 3.
Difference of odd and even place digits $= 6 - 6 = 0$, so it is divisible by 11.
$111,111,111,111 = 111 \times 1,001,001,001 = 3 \times 37 \times 111 \times \text{etc.}$
Checking factorizations: $111 = 3 \times 37$, $1001 = 7 \times 11 \times 13$.
Thus, it is divisible by $3, 11, 37, 111,$ and $1001$.
Correct Option: (d) 3, 11, 37, 111 and 1001
Which of the following numbers is not divisible by 18?
Rule: $18 = 2 \times 9$. To be divisible by 18, a number must be even (ending in an even digit) and divisible by 9.
$65043$ ends in $3$ (odd), so it cannot be divided by 2 or 18.
Correct Option: (d) 65043
The number 89715938* is divisible by 4. The unknown non-zero digit marked as * will be
Rule: A number is divisible by 4 if its last two digits form a number divisible by 4.
Last two digits: $8*$.
$82, 84, 86, 88$ are options for $8*$. Given choices: 2, 3, 4, 6.
Non-zero digit options given are 2, 4, 6. Testing $84 \div 4 = 21$.
Correct Option: (c) 4
A number is divisible by 11 if the difference between the sums of the digits in odd and even places respectively is
Definition/Rule: A number is divisible by 11 if the difference between the sums of the digits in odd and even places is zero or a multiple of 11.
Correct Option: (d) zero or a multiple of 11
Which one of the following numbers is divisible by 11?
Test options using 11's rule:
(d) $8432718 \rightarrow$ Odd positions: $8+7+3+8 = 26$. Even positions: $1+2+4 = 7$.
Difference $= 26 - 7 = 19$ (Not divisible).
(a) $4823718 \rightarrow$ Odd: $8+7+2+4 = 21$. Even: $1+3+8 = 12$. Difference $= 9$.
(c) $8423718 \rightarrow$ Odd: $8+7+2+8 = 25$. Even: $1+3+4 = 8$. Difference $= 17$.
(b) $4832718 \rightarrow$ Odd: $8+7+3+4 = 22$. Even: $1+2+8 = 11$. Difference $= 22 - 11 = 11$ (Divisible).
Correct Option: (b) 4832718
7386038 is divisible by
Number: $7386038$.
Sum of digits $= 7+3+8+6+0+3+8 = 35$ (Not divisible by 3 or 9).
Last two digits $= 38$ (Not divisible by 4).
Alternating sum for 11: $(8+0+8+7) - (3+6+3) = 23 - 12 = 11$ (Divisible by 11).
Correct Option: (d) 11
Which of the following numbers is a multiple of 8?
Rule: A number is divisible by 8 if its last three digits are divisible by 8.
Checking last three digits:
(a) $872 \div 8 = 109$ (Exactly divisible).
Correct Option: (a) 923872
If m and n are integers divisible by 5, which of the following is not necessarily true?
Let $m = 5$ and $n = 5$ (both divisible by 5).
$m + n = 10$, which is divisible by 10.
However, if $m = 10$ and $n = 5$, $m + n = 15$, which is not divisible by 10.
Therefore, statement (a) is not necessarily true.
Correct Option: (a) m + n is divisible by 10