Clocks & Time: 8 Important Shortcuts Explained

We need to get couple of basic facts clear:

* Speed of the hour hand = 0.5 degrees per minute (dpm) {The hour hand completes a full circle or 360 degrees in 12 hours or 720 minutes}

* Speed of the minute hand = 6 dpm {The minute hand completes a full circle in 60 minutes}

* At ‘n’ o’ clock, the angle of the hour hand from the vertical is 30n

Clock problems can be broadly classified in two categories:

a) Problems on angles

b) Problems on incorrect clocks

c) Problems on angles

Finding the angle between the hands category clock problems  major questions which is quite time taking and difficult to solve.

Some easy tricks to solve

TYPES OF NUMBERS

I. Natural Numbers : Counting numbers I, 2. 3, 4, 5, ..... are called natural artillery.

2. Whole Numbers : All counting numbers together with zero form the set of whole number. Thus,

   (A) 0 is the only whole number which is not a natural number.

   (B) Every natural number is a whole number

3. Integers : All natural numbers. 0 and negatives of counting numbers ie., - 3, - 2, - 1. 0. 

I. 2, 3...... together form the set of integers

(A) Positive Integers t 11. 2. 3, 4, .....I is the set of all positive integers.

(B) Negative Integers : (- I, - 2, - 3......I is the set of all negative integers.

(C) Non-Positive and Non-Negative integers : 0 is neither positive nor negative. 

    So, 0, 1, 2, 3. ..... I represents the set of non-negative integers, while (0, - I, - 2, - 3,  ) 

    represents the set of non-positive integers.

TESTS OF DIVISIBILITY OF NUMBERS (ShortCuts)

I. Divisibility By 2 : A number is divisible by 2. if its unit's digit is any of 0, 2. 4. 6. 8.Ex. 84032 is divisible by 2. while 05935 is not.

Consider a two digit number (10*a+b). Factor this to 2*5*a + b. This shows that all numbers ending a zero are divisible by two, so if the ones digit is divisible by two, the entire number is as well.

2. Divisibility By 3 : A number is divisible by 3, if the sum of its digits is divisible by 3.
Consider a 2 digit number 10*a + b = 9*a + (a+b). 
We know that 9*a is divisible by 3, so 10*a + b will be divisible by 3 if and only if a+b is. 
Similarly, 100*a + 10*b + c = 99*a + 9*b + (a + b + c), and 99*a + 9*b is divisible by 3, so the total will be divisible by 3 if and only if a + b + c is. 

Ex. 592482 is divisible by 3, since sum of its digits = (5 + 9 + 2 +4 + 8 + 2) = 30, which is divisible by 3. But, 804329 is not divisible by 3, since sum of its digits • (8 +0+ 4 + 3 + 2 + 9)= 32. which is not divisible by 3.

3. Divisibility By 4 : A number is divisible by 4, if the number formed by the last two digits is divisible by 4.
Consider a three digit number (100*a + 10*b + c)
Factor the first digit: 4*25*a + 10*b + c 

Ex. 892648 is divisible by 4, since the number formed by the last two digits is 48. which is divisible by 4. But, 749282 is not divisible by 4. since the number formed by the last two digits is 82, which is not divisible by 4.

H.C.F and L.C.M of Numbers (Shortcuts)

I. Factors and Multiples : If a number a divides another number b exactly, we say that a is a factor of b.  In this case, b is called a multiple of a

II. Highest Common Factor (H.C.F.) or Greatest Common Measure(G.C.M.) or Greatest Common Divisor (G.C.D.)  The H.CF of two or more than two numbers is the greatest number that divides each of them exactly.
There are two methods of finding tho H.C.F of a given set of numbers

1.    Factorization Method : EXpress each one of the given numbers as the product of prime factors
 The product of least powers of common prime factors gives H.C.F

2.    Division Method: Suppose we have to find the H.C.F of two given numbers. Divide the larger number  by the smaller one. Now divide the divisor by the remainder Ripeat the process of dividing the preceding number  by the remainder last obtained till zero is obtained as remainder. The lastt divisor is the required H.C.F. 
Finding the H.C.F. of more than two numbers : Suppose we have to find the H.C.F. of three numbers then H.C.F. of ( H.C.F of any two) and (the third number ) gives the H C F of three given numbers.  Similarly the H.C.F of more than three numbers may be obtained.