Know the DNA of Divisibility, Exponent & divisibility, divisibility rules,Multiples concepts for GRE
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| Nilenodhigherstudy |
Know the DNA of Divisibility for
We all know what means divisibility, let's understand the following concept so that you can solve every prompt of divisibility.
1. If A×B=C,
then we say A & B are factors of C. To say A is a factor of C is to say
that we can multiply A by some integer & the product will equal C. For
example, 3 is a factor of 6, or 25 is a factor of 100.
Remember, 1 is a factor of every positive integer.
Every integer is a factor of itself. Every positive integer greater than 1 has
at least 2 factors: 1 & itself.
2. If C÷A=B,
then we say A is a divisor of C because it divides evenly into C. One number
divides evenly into another when the quotient is an integer. We can also say
that C is divisible by A. There is no difference between factor & divisor that
means, these two words mean exactly the same thing. So, GRE may interchangeably
test following 3 ways:
1) 8 is a factor of 24
2) 8 is a divisor of 24
3) 24 is divisible by 8
Same way,
8 is not a factor of 12
8 is not a divisor of
12
12 is not divisible by
8
Because the quotient 12÷8
is not an integer & what amounts to the same thing, there is no positive
integer B such that the product 8×B
equals 12. So, what happens when we
divide 12 by 8? Mathematically two things happen,
1) If two numbers
divide evenly, then (integer) divided by (integer) equals (integer): the quotient is an integer, & that's it.
2) If two items don't
divide evenly, we have two possible options, each perfectly correct.
Option#1: Have an
integer quotient & an integer remainder, such as 8 go into 12 once, with the remainder of 4.
Option#2: Express
quotient as a fraction or a decimal, Such as 12/8=3/2=1 & 1/2=1.5
Don’t be confused, GRE
will indicate which one should use
Lets’ answer some
questions,
How would we find all
the positive factors of 36?
Methods: Apply a list
factor pairs: the pairs of numbers that have a product of 36.
1, 36
2, 18
3, 12
4, 9
6, 6----stop here,
because two factors are the same. 1st factor will never be bigger than 2nd
factor.
Negative integers:
Technically, it is true that
Both +4 & -4 are
factors of -12.
Both +4 & -4 are
divisors of -12
-12 is divisible by
both +4 & -4
Exponent
& divisibility
You have to understand the following concept to perform better in Exponent & divisibility-based math.
1) Case-1: (an-bn) is always
divisible by (a-b). For example, 52-32=16 here, a-b=5-3=2 & 16 is divisible by 2. Again, 53-33=98 here
a-b=2 & 98 is divisible by 2. So, 43-23=56 here, a-b=2 & 56 is divisible by 2
2) Case-2: (an-bn) is divisible by (a+b) if n is even. E.g. 42-22=12 here, a+b=6 & 12 is divisible by 6. But if n=odd, then it will not divisible.
3) Case-3:(an+bn) is divisible by (a+b) if n is odd & not divisible by (a+b) if n is even. E.g. 53+23=133 here, a+b=7 & 133 is divisible by 7. 52+22=29 & not divisible by 7.
Know the divisibility rules
Divisibility Rule for
2: If the unit or last digit is even,
then the number will be even & divisible by 2. [All the even numbers are
divisible by 2]
Divisibility Rule for
3: If the sum of the number is divisible by 3 then the number is divisible by
3.
Divisibility Rule for
9: If the sum of the number is divisible by 9 then the number is divisible by
9. [trick to remember 3, 9=sum]
Divisibility Rule for 6: If the number divisible by 2 & 3 then it will divisible by 6
[trick to remember 6=2, 3]
Divisibility Rule for 4:
If the last two digits are divisible by 4 then the entire number is divisible
by 4
Divisibility Rule for 8: If the last three digits are divisible by 8 then the entire number is divisible by 8
[trick to remember 4, 8=2, 3 d, that means 4 & 8 are divisible by 2nd & third number] or if the number is divisible by 2 in 3 succession time then that number also be divisible by 8.
Divisibility Rule for 12: If the number is divisible by both 3 & 4, then it will be divisible by 12.
[trick to remember 12= 3, 4 that means 12 is divisible by 3 & 4, 34]
Divisibility Rule for 5: If unit or last digit is 5 or 0, then the number will be divisible by 5
[trick to remember 5=5, 0 or use any self develop trick to remember it]
Divisibility Rule for 10: If unit digit is 0, then the number will be divisible by 10
[trick to remember 10=0]
Divisibility Rule for 25: If unit digit is 25, 50, 75, 00, and then the number will be divisible by 25.
[trick to remember 25=3 multiple, 00]
Divisibility Rule for
7: 897, double last number that means, 7, then divide from rest of the number i.e. 89.
So, 89-14= 75, is 75 divided by 7? If the answer is yes, then this number will
divisible by 7, if not then not divisible by 7.
Divisibility Rule for
11: 93247296121,
Step#1: Add odd place number=9+2+7+9+1+1=29
Step#2: Then add even
place number=3+4+2+6+2=17
Step#3: Find out
difference=29-17=12
Step#4: Are 12
divisible by 11? No, this number will not divisible by 11.
Divisibility Rule for
13:
Step#1: Find out last digit
Step#2: Then multiply
last digit by 4 & add with rest of the number.
Step#3: Follow same
procedure until an easy numbers comes.
Apply-1:
Are 702 divisible by 13?
Step#1: last digit=2
Step#2: 2×4
+ 70 (70= rest of the number) =78 (step#3 is followed to get an easy number. 39
is comparatively easy than 78)
Step#3: 8×4+7=39
Is 39 divisible by 13, yes, so the number is divisible by 13.
Apply-2:
Are 3133 divisible by 13?
Step#1: last digit=3
Step#2: 3×4
+ 313 (313= rest of the number) =325 (step#3 is followed to get an easy number.
Step#3: 5×4+32=52 If you think 52 is hard to think then follow
another step.
Step#4: 2×4+5=13
Is 13 divisible by 13, yes, so the number is divisible by 13.
Apply-3:
Are 11102 divisible by 13?
Step#1: last digit=2
Step#2: 2×4
+ 1110 (1110= rest of the number) =1118 (step#3 is followed to get an easy
number.
Step#3: 8×4+111=143 If you think 143 is hard to think then
follow another step.
Step#4: 3×4+14=26
Is 26 divisible by 13, yes, so the number is divisible by 13.
Divisibility Rule for 17:
Same as 13 with slight difference. Take 5 instead of 4, do subtract instead of
addition.
Apply-1: 1054 last digit=4, 4×5,
rest of the number=105, so 105 – 20=85. Follow another steps, 5×5
-8=-17, take 17 neglecting minus sign. Are 17 divisible by 17, yes, so the
whole number divisible by 17.
Divisibility Rule for
19: Double last digit & then add with rest of the number. If the new number
is divisible by 19, then the whole number will be divisible by 19.
Apply-1: is 2375
divisible by 19 ?
Last digit double= 5×2
Rest of the number= 237
New number= 247, if
this numbers seems hard for you then follow repeated steps.
Last digit double= 7×2
Rest of the number=24
New number=38, so, this
is comparatively an easy number & is divisible by 19.
Multiples concepts for GRE
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| Nilenodhigherstudy |
In an above factor
tree, below are factors & above are multiples, for example, factors of 12 are
6,2,3,2 all are not greater than 12. And multiples are 24, 36, 48 all are
greater than 12.
Keynotes:
(1) If X is
a multiple of Y, then Y is a factor of X. i.e. 36 is a multiple of 12; 12 is a factor of 36.
(2) If X is a multiple
of Y, then X is divisible by Y, for example, 36 is a multiple of 12; 36 is divisible by
12
(3) A multiple of a number can be obtained by multiplying that number by an integer.
Multiple relationships:
(1) If you add two multiples of a number together, the sum will itself be multiple. For example, 24+36=60, here, 24 is a multiple of 12 & 36 is also a
multiple of 36.
(2) If you subtract two
multiples of a number, the difference will itself be multiple. For example, 48-24=24,
here, 24 is a multiple of 12.
(3) If you multiply two
multiples of a number together, the product will itself be multiple. For example, 24×36=864,
here, 864 is a multiple of 12.
Remember, this is not
applicable in division
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