FACTORS AND MULTIPLES
When any number can divide another number then the said number is counted as ‘Divisor’ and this divisor is called as “Factor” of the ‘Dividend’, and the ‘Dividend’ is called “Multiple”
Some Important fact about Factor & Multiples –
Eternal Fact of Factor –
a) 1 is a factor of every and any Number
b) Every number is a factor of itself
c) A factor of a number is less than or equal to the number
d) 1 is the only number having one factor, namely itself
e) Every number other than 1 has at least two factors, 1 and itself.
Eternal Fact of Multiplies –
1) Each and every number is multiply of 1, example- 25 = 25 x 1 , 35 = 35 x 1, 40= 40 x 1
2) Every number is multiple of its own, example- 12 = 12 x 1, 24 = 24 x 1, 120 = 120 x 1
3) Every number has infinitely many multiples as shown above.
4) Every multiple of a number is greater than or equal to the number .
Factors & Multiples
As we all know that, ‘Divisor’ is called as Factor and ‘Dividend’ is called Multiple, so for an example –
If, 25 is divided by 5, then 5 is Factor and 25 is multiple of 5.
Similarly, If 44 is divided by 11, then 11 is Factor and 44 is multiple of 11.
Similarly, If 108 is divided by 12, then 12 is Factor and 108 is multiple of 12.
Similarly, If 96 is divided by 8, then 8 is Factor and 96 is multiple of 8.
Similarly, If 56 is divided by 7, then 7 is Factor and 56 is multiple of 7.
DIVISIBILITY
As we all know that, 12, 24, 6, 28, 18,……. is Divisible by 2, and all the even numbers are divisible by 2 and 237, 411, 235, 87, 91,……… are not divisible by 2 and all the odd number is not divisible by 2.
But on the other way we can describe in that way, if the sum of the digit of the given number is divided by 2, then the number must be divided by 2.
For Example- If we consider 1346 is to be divide by 2, then as per the given condition we have to add the digit of the given number- (1+3+4+6) = 14 and 14 ÷ 2 = 7, so 1346 is divisible by 2 as 1346 ÷ 2 = 673
if we consider 3476 is to be divide by 2, then as per given condition we have to add the digit of given number- (3+4+7+6) = 20 and 20 ÷ 2 = 10, so 3476 is divisible by 2 as 3476 ÷ 2 = 1738
For Example- If we consider 3699 is to be divide by 3, then as per the given condition we have to add the digit of the given number- (3+6+9+9) = 27 and 27 ÷ 3 = 9, so 3699 is divisible by 3 as 3699 ÷ 3 = 1233
If we consider 2345 is to be divide by 3, then as per the given condition we have to add the digit of the given number- (2+3+4+5) = 14 and 14 ÷ 3 = not divisible, so 2345 is not divisible by 3 as 2345 ÷ 3 = not divisible
For Example- if we consider 96984 is to be divide by 4, then as per the given condition we have to add the digit of the given number- (9+6+9+8+4) = 36 and 36 ÷ 4 = 9, so 96984 is divisible by 4 as 96984 ÷ 4 = 24246.
Or
A number is divisible by 4 only when the number formed by its last 2 digits (One’s place & Ten’s place digit) is divisible by 4. That is, the last two digits of the given number 96984 is 84, and 84 is divisible by 4, hence 84 ÷ 4 = 21. So, 96984 is divisible by 4, hence 96984 ÷ 4 = 24246.
For Example- A number is divisible by 5 only when it’s unit digit is 0 or 5. If we consider the number 2355 then we can observe that last digit of the given number is 5, so the given number is divisible by 5. hence 2355 ÷ 5 = 471
Or
If 2355 is to be divide by 5, then as per the given condition we have to add the digit of given number- (2+3+5+5) = 15 and 15 ÷ 5 = 3, so 2355 is divisible by 5, hence as 2355 ÷ 5 = 471.
For Example- If any number is divisible by 6 only when it is divisible by both 2 & 3. Consider 5382 is the number to be divided by 6. where 2 is the last digit of the given number, so it's divisible by 2 on the other way sum of the digits of given number is 5+3+8+2 = 18 and 18 is divisible by 3, so 18 ÷ 3 = 6, thus the given number divisible by both the number 2 & 3, 5382 is divisible by 6, hence 5382 ÷ 6 = 897.
For Example- Any number is divisible by 7 only when, if the difference between double the digit at Ones place and the number formed by rest of its digits is divisible by 7.
For Example – Consider 16555 is the number to be divide by 7. As per the condition last digit (One’s place) of the given number is 5 and is to be converted in double-digit, i.e. 5 x 2 = 10, and now 16555 => { 1655 – (5 x 2)} = 1645, now we have to check whether 1645 is divisible by 7 or not. 1645 ÷ 7 = 235
For Example - Any number is divisible by 8 only when, if the number formed by it’s last 3 digits on it’s extreme right is divisible by 8. If we consider the number 25216, then the last three digits of the given number is 216, then we will try to divide 216 by 8 and it has been found that 216 is divisible by, hence 216 ÷ 8 = 27, so we can conclude that, the given number 25216 is divisible by 8 and hence 25216 ÷ 8 = 3152.
For Example- Any number is divisible by 9 only when, if the sum of its digits of the given number is divisible by 9.
If we consider the number 25875, then the sum of the digits of the given number is 2+5+8+7+5 = 27. now we have to check whether 27 is divisible by 9 or not and we find that, 27 is divisible by 9 and hence 27 ÷ 9 = 3. So, hence we can conclude that 25875 is divisible by 9, 25875 ÷ 9 = 2875.
For Example- Any number is divisible by 10 only when, if the last digit of the given number is 0 (Zero). If we consider the number 25470 then we have to check whether the given number is divisible by 10 or not and we can find that, the given number is divisible by 10, hence 25470 ÷ 10 = 2547.
So, the given numbers like 3400 , 750, 3580, 125000 all are divisible by 10