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Numbers
1.

12112 को 11 से और 13223 को 13 से भाग देने पर प्राप्त शेषफलों का योग है - 

Answer:Option 1

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2.Find the remainder when 73 × 75 × 78 × 57 × 197 × 37 is divided by 34.

Answer:Option 1

Explanation:

Solution:
Remainder,
73×75×78×57×197×3734
      ⇒
5×7×10×23×27×334

[We have taken individual remainder, which means if 73 is divided by 34 individually, it will give remainder 5, 75 divided 34 gives remainder 7 and so on.]

5×7×10×23×27×334
     ⇒
35×30×23×2734
   [Number Multiplied]
35×30×23×2734
    ⇒
1×4×11×734


[We have taken here negative as well as positive remainder at the same time. When 30 divided by 34 it will give either positive remainder 30 or negative remainder -4. We can use any one of negative or positive remainder at any time.]

1×4×11×734
    ⇒
28×1134
  ⇒
6×1134
  ⇒
6634
 ⇒
R
⇒ 32
Required remainder = 32.

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3.Find the remainder when 6799 is divided by 7.

Answer:Option 1

Explanation:

Solution:
Remainderof67997==R(63+4)997[63isdivisibleby7foranypower,sorequiredremainderwilldependonthepoerof4]Requiredremainder:4997==R==4(96+3)7437647(63+1)7==R1Note:47remainder=4(4×4)7=167remainder=2(4×4×4)7=647=1(4×4×4×4)7=2567remainder=4(4×4×4×4×4)7=2

If we check for more power we will find that the remainder start repeating themselves as 4, 2, 1, 4, 2, 1 and so on. So when we get A number having greater power and to be divided by the other number B, we will break power in (4n+x) and the final remainder will depend on x i.e. Ax/B.

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4.Let N = 1421 × 1423 × 1425. What is the remainder when N is divided by 12?

Answer:Option 1

Explanation:

Solution:
Remainder,
(1421*1423*1425)/12 ==R==>(5*7*9)/12

[Here, we have taken individual remainder such as 1421 divided by 12 gives remainder 5, 1423 and 1425 gives the remainder as 7 and 9 on dividing by 12.]

Now, the sum is reduced to,
(5*7*9)/12 = (35*9)/12
(35*9)/12 = Remainder ==> -1 * -3 = 3 [Here, we have taken negative reminder.] So, required remainder will be 3.

Note: When, 9/12 it gives positive remainder as 9 and it also give a negative reminder -3. As per our convenience,we can take any time positive or negative reminder.

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5.What is the least number of soldiers that can be drawn up in troops of 12, 15, 18 and 20 soldiers and also in form of a solid square?

Answer:Option 1

Explanation:

Solution:
In this type of question, We need to find out the LCM of the given numbers.
LCM of 12, 15, 18 and 20;
12=2×2×3;15=3×5;18=2×3×3;20=2×2×5;

Hence, LCM =
2×2×3×5×3

Since, the soldiers are in the form of a solid square.
Hence, LCM must be a perfect square. To make the LCM a perfect square, We have to multiply it by 5, hence, the required number of soldiers =
2×2×3×3×5×5
    = 900.

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6.Find the least number which will leaves remainder 5 when divided by 8, 12, 16 and 20.

Answer:Option 1

Explanation:

Solution:
We have to find the Least number, therefore we find out the LCM of 8, 12, 16 and 20.
8 = 2*2*2;
12 = 2*2*3;
16 = 2*2*2*2;
20 = 2*2*5;
LCM = 2*2*2*2*3*5 = 240;
This is the least number which is exactly divisible by 8, 12, 16 and 20.
Thus,
required number which leaves remainder 5 is,
240+5 = 245.

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7.76n- 66n, where n is an integer >0, is divisible by

Answer:Option 1

Explanation:

Solution:
76n66n=7666=(73)2(63)2=(7363)(73+63)=(343216)×(343+216)=127×559=127×13×43

Clearly, it is divisible by 127, 13 as well as 559.

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8.After the division of a number successively by 3, 4 and 7, the remainder obtained is 2, 1 and 4 respectively. What will be remainder if 84 divide the same number?

Answer:Option 1

Explanation:

Solution:
As the Number gives a remainder of 4 when it is divided by 7, then the number must be in form of (7x +4).

The same gives remainder 1 when it is divided 4, so the number must be in the form of {4*(7x+4)+1}.

Also, the number when divided by 3 gives remainder 2, thus number must be in form of [3*{4*(7x +4)+1}+2]

Now, On simplifying,
[3*{4*(7x+4)+1}+2]
= 84x+53.
We get the final number 53 more than a multiple of 84. Hence, if the number is divided by 84,
the remainder will be 53.

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9.Find the remainder when 496 is divided by 6.

Answer:Option 1

Explanation:

Solution:
4966,wecanwriteitinthisform(62)966Now,Remainderwilldependonlythepowersof - 2.So,(2)966,itissameas([2]4)246,itissameas(16)246Now,(16×16×16×16......24times)6Ondividingindividually16wealwaysgetaremainder4.So,(4×4×4×4......24times)6Hence,RequiredRemainder=4

NOTE: When 4 has even number of powers, it will always give remainder 4 on dividing by 6.

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10.What is the sum of all two digit numbers that gives a remainder of 3 when they are divided by 7?

Answer:Option 1

Explanation:

Solution:
The two digit number which gives a remainder of 3 when divided by 7 are:
10, 17, 24 ..... 94.
Now, these number are in AP series with
1st Term, a = 10;
Number of Terms, n = 13;
Last term, L = 94 and
Common Difference, d = 7.
Sum,
={n×a+L2}=13×52=676

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