Arithmetic aptitude :: Numbers :: General Questions

• Numbers - General Questions
• Numbers - Important Formulas and Facts

Divisibility Rule

A positive integer N is Divisible by,

2: if the last digit of N is 2, 4, 6, 8, or 0;
3: if the sum of digits of N is a multiple of 3;
4: if the last 2 digits of N are a multiple of 4;
5: if the last digit of N is either 0 or 5;
6: if N is divisible by both 2 and 3;
7: if subtracting twice the last digit of N from the remaining digits gives a multiple of 7
(e.g 658 = 65 - 2 x 8 = 65 - 16 = 49, which is a multiple of 7)
8: if the last 3 digits of N are a multiple of 8;
9: if the sum of digits of N is a multiple of 9;
10: if the last digit of N is 0;
11: if the difference of the alternating sum of digits of N is a multiple of 11
(e.g. 510510 = (5 + 0 + 1) - (1 + 5 + 0) = 6 - 6 = 0, So this is divisible by 11)
12: if N is divisible by both 3 and 4.

For more informations about Divisibility rules, please check, Wikipedia

Unit Digit

Unity means one, so the units digit is the digit in the one's column.
(e.g for the number 168, 8 is the unit digit)

To find out the unit digit in the product of the numbers, only consider the unit digit in each numbers

e.g. 753 x 43 x 1236 x 864

unit digit in each numbers

3 x 3 x 6 x 4 = 216, So 6 is the unit digit.

(51 + 52 + 53 + ..... + 100) is equal to :

Answer:

Explanation:

This is an A.P. in which a = 51, I = 100 and n = 50.
$\frac{n}{2}(a+I)=\frac{50}{2}(51+100)$
= (25 x 151) = 3775.

The sum of first 45 natural numbers is :

Answer:

Explanation:

Let S_n =(1 + 2 + 3 + ... + 45). This is an A.P. in which a =1, d =1, n = 45.

$S_{n}= \frac{n(n+1)}{2}=\frac{45(45+1)}{2}$

=1035

186 x 186 + 159 x 159 - 2 x 186 x 159 = ?

Answer:

Explanation:

The order of operations is:
Parentheses means brackets()
Exponents (and Roots) means power
Multiplication & Division
Addition & Subtraction

(186 x 186) + (159 x 159) - (2 x 186 x 159) = 729

The digit in the unit's place of the number represented by (7⁹⁵ - 3⁵⁸) is :

Answer:

Explanation:

If you look into the power of 7 then you will notice a sequence:

7¹=7,7²=49, 7³=343, 7⁴=2401, 7⁵=16807

And the unit digits keep on repeating as { 7, 9, 3, 1 } It repeats after every 4 powers ,

So divide 95 by 4 You will get 3 as the remainder . The third number in the series is 3. This implies the unit digit of 7⁹⁵ is 3 .

Similarly, do for 3⁵⁸

3¹=3, 3²=9, 3³=27, 3⁴=81, 3⁵=243 And the unit digits keep on repeating as { 3, 9, 7, 1 } Now divide 58 by 3 and get the remainder as 2 Second number in the series is 9.

This implies unit digit of 3⁵⁸ is 9

Unit digit of 7⁹⁵−3⁵⁸ = 3−9 = 13-9 (borrow from the previous digit)

= 4

The unit's digit in the product (3127)¹⁷³ is :

Answer:

Explanation:

Unit digit in (3127)¹⁷³ = Unit digit in (7)¹⁷³.

Now, 7⁴ gives unit digit 1.

(7)¹⁷³= (7⁴)⁴³ * 7¹. Thus, (7)¹⁷³ gives unit digit 7.

• Numbers - General Questions