Divisibility and Re...

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# Divisibility and Remainders

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Topic starter

Find the maximum value of n such that 50! is perfectly divisible by 12600 raised to the power n.

10?

12600= (2^3)* (3^2)*(5^2)*(7)

powers of 2 in 50! by successive division = 50+25+12+6+3+1= 97, hence power of 2^3 will be 97/3 = 32
same way powers of 3 = 16+5+1 = 22; powers of 3^2 = 11
powers of 5 = 10+2 = 12; powers of 5^2 = 6
powers of 7 = 7+1 = 8
from the above set since there are only 6 powers of 5^2 are available, hence maximum value of n can be 6 only

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Hello Apoorva,

Find the solution below, If you wants more explanation of this question then ask for detail explanation.

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What is the remainder when 7^2017 is divided by 33?

Can you explain it without the use of Chinese Remainder Theorem.

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Yes Apoorva,

Find the solution without the use of Chinese Remainder Theorem

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Topic starter

What is the remainder of 38! / 41 ?

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Hello ritika,

Using Wilson's theorem, we get
(41 - 1)! = 40! = -1mod41 = 40mod41
=> 39! = 1mod41
and let 38! = xmod41
=> 39x = 1mod41
=> -2x = 1mod41
=> 41y - 2x = 1
=> x = 20 and y = 1.
So Remainder when 38! is divided by 41 is 20.

Thanks sir

Sir,

I cant understand the step when

-2x=1mod41

Can you explain last three steps again

Hello Apoorva,

Find the solution in detail

As per Wilsons theorem when (n - 1)! is divided by p we will get a remainder of - 1 when p is a prime number.

As per the theorem when 40! is divided by 41 we will get a remainder of – 1

When (41 -1)! Is divided by 41, remainder is 1.

It can be written as 40 X 39 X 38! divided by 41 will give a remainder of -1

(41–1)(41–2) 38! / 41 will give a remainder of - 1 since 41 in the bracket will get divided by 41 we can write as -1*-2**Y /41 will give a remainder of -1 where Y is 38!

Rem 2Y / 41 = -1

or Rem 2Y = 41a - 1 where a is the constant factor

since the left hand side is a multiple of 2 the right hand side also will have to be multiple of 2. we will have to substitute values for a in such a way that 41a - 1 will is a multiple of 2.

Put a =1

2Y = 41 x 1 - 1

2Y =40 => Y=20

Where Y is the remainder when 38! Is divided by 41

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Topic starter

If a number N, when divided by D, gives remainder 23. The number, when divided by 12D, gives remainder 104. What will be the remainder when the number is divided by 6D?

Hi Ritika,

N = 23modD
N = 104mod12D
=> 104 = 23modD
=> 81 = 0modD
=> D = 27 or 81
Now Nmod6D = 104mod6D [Because 12d is divisible by 6D]
And 27*6 = 162<104.
=> N = 104mod6D

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Topic starter

Find the remainder when 199! is divided by 67^3 ?

Hello , Ritika

199 ! = 1× 2 × 3 × ......67 × 68 × 69 ×........132 × 133×134×......199.

1× 2 × 3 × ......67 × 68 × 69 ×........132 × 133×134×......199 mod 67³

1×2×3×......66×68×69×.....133×2×135×.......199 mod 67

66! × 66! × 65! × 2 mod 67

We know , ( P-1)! mod P = ( P-1 ) or -1 [ Wilson's Theorem ]

-1 × -1 × 1× 2= 2

Hence , net remainder 2× 67² = 8978

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Find the remainder when 1021^1022 is divided by 1023 ?

Hello Abhi ,

1021^1022 mod 1023

Using Negative remainder concept

(-2)^1022 mod 1023

2^1023 mod 1023

2^10 mod 1023 = 1024 mod 1023 = 1

so (2^10)^102 * 2^2  mod 1023 = 4

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1234123412341234.........upto 400 digits/999, then what is the remainder?

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As 1000 = 103 ≡ 1 mod999.

So 12341234.... 400 digits ≡ 1 + 234 + 33(123 + 412 + 341 + 234) mod999 ≡ 1 + 234 + 33(111) mod999 ≡ 901 mod999

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what is the remainder when    123123123............(300 digits)  is divided by 504

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hi vidisha ,

there is another topic by the name of remaineder plz post your question under that topic

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I have a doubts in 23 and 24 th question

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Find the remainder when 7^2008 +9^2008 is divided by 64?

Hello Nishant .

Do you have any idea about binomial expansions   ?

Approach : Using binomial expansions .

If you have any doubts  feel free to write back

7^2008 + 9^2008 mod 64

( 8 -1) ^2008 + (8 +1)^9 ^2008 mod 64

Now the binomial expansion of ( 8 -1)^2008 = 2008C1 × 8^0 × (1)^2008 + 2008C1 × 8 × (-1)^2007 + a multiple of 64

and the binomial expansion of (8 + 1)^2009 = 2008C1 × 8^0 × (1)^2008 + 2008C1 × 8 × (+1)^2007 + a multiple of 64

Hence ( 7 - 1)^2009 + ( 7 +1)^2009 mod 64

= 2008C1 × 8^0 × (1)^2008 + 2008C1 × 8 × (-1)^2007 + 2008C1 × 8^0 × (1)^2008 + 2008C1 × 8 × (1)^2007 mod 64

1 + 1 mod 64 = 2 .

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Hello Sir

Question 1- What is the remainder when f(a) = a^15+a^14+a^13+........a^2+a is divided by a^2-1

A. 7a+8

B. 8a +7

C. 9a+6

D. 5a+8

Hi Mayank !

We know, the remainder when f(x) = a + bc + cx^2 + dx^3 +...... is divided by  ( x - a ) is f(a).

( Check NS - 4 Class Sheet )

Hence , Substitute a^2 = 1 in the expression  a^15+a^14+a^13+........a^2+a to get the remainder .

(a^2)^7. a + (a^7)^2 + ( a^2)^6 . a + (a^2)^6 + ...... + (a^2)a + a + 1

(1)^7. a + (1)^2 + (1)^6 . a + (1)^6 + ...... + (1)a + a + 1

= 8a + 7.

Option B.

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