Number System - Aptitude MCQ Questions and Solutions with Explanations

Divisibility law of 4: A number divisible by 4 if its last 2 digit is divisible by 4
Divisibility law of 11: If the difference of the alternating sum of digits of the number is a multiple of 11 (e.g. 2343 is divisible by 11 because 2 - 3 + 4 - 3 = 0, which is a multiple of 11).
⇒ 15x1y2 is divisible by 4 if y = 1 or 3
If y = 1, then
⇒ 15x112 is divisible by 11 if x = 6
If y = 3 then
⇒ 15x132 is divisible by 11 if x = 4
If y = 1 and x = 6, then
⇒ x + y = 1 + 6 = 7
If y = 3 and x = 4, then
⇒ x + y = 3 + 4 = 7

On arranging the digit of number 150328 in ascending = 853210
According to the question,
853210 - 13 × 5 = 853210 - 65 = 853145
Clearly it is divisible by 5.

10 digit number 1330x558y2 is divisible by 88, means number is also divisible by 11 and 8
⇒ Sum of digit of the number even place = 3 + 0 + 5 + 8 + 2 = 18
⇒ Sum of digit of the number odd place = 1 + 3 + x + 5 + y = 9 + x + y
As we know,
⇒ 9 + (x + y) – 18 = 0
⇒ x + y = 9

$$\eqalign{ & 2\frac{1}{2} + 3\frac{1}{3} + 4\frac{1}{4} + 5\frac{1}{5} \cr & = \left( {2 + 3 + 4 + 5} \right) + \left( {\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5}} \right) \cr & = 14 + \left( {\frac{{30 + 20 + 15 + 12}}{{60}}} \right) \cr & = 14 + \frac{{77}}{{60}} \cr & = 14 + 1 + \frac{{17}}{{60}} \cr & = 15 + \frac{{17}}{{60}} \cr & \therefore {\text{Required answer}} = 1 - \frac{{17}}{{60}} \cr & = \frac{{60 - 17}}{{60}} \cr & = \frac{{43}}{{60}} \cr} $$

$$\eqalign{ & {\text{Let number are }}x{\text{ and }}y \cr & {\text{According to the question,}} \cr & x \times y = 48 \cr & {\text{and }}x = \frac{{2 + \mathop {\mathop 5\limits^ \uparrow }\limits^{{\text{total fingers in one hand}}} \times 2}}{3} = \frac{{12}}{3} = 4 \cr & y = \frac{{48}}{4} = 12 \cr} $$

According to the question,
$$\eqalign{ & \,5\,\,\,\,\,E\,\,\,\,\,9 \cr & \,2\,\,\,\,\,F\,\,\,\,\,8 \cr & \,3\,\,\,\,\,G\,\,\,\,\,7 \cr & \underline {\overline {11\,\,\,\,\,1\,\,\,\,\,4\,} } \cr} $$
⇒ Sum of 2, E, F and G must be 11. For maximum F we will have to take E and G zero.
∴ F = 9

(10x + y)(x + y) = 424 . . . . . . (i)
(10y + x)(x + y) = 280 . . . . . . (ii)
Add equation (i) and (ii)
11(x + y)(x + y) = 704
(x + y)² = 64
x + y = 8

In this type of questions go through option. We take option B.
$$\frac{{34}}{{13}}$$ Remainder is 8.
$$\frac{{34}}{7}$$ Remainder is 6.
It satisfy the condition so answer in option B.

If 8764x5, is divisible by 9 then, sum of digit
8 + 7 + 6 + 4 + x + 5 = 0/9/18/36/45
30 + x = 45
x = 15

Sum of first $$n$$ natural numbers $$\frac{{n\left( {n + 1} \right)}}{2}$$
∴ Required average $$ = \frac{{n\left( {n + 1} \right)}}{{2 \times n}} = \frac{{n + 1}}{2}$$