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1. Find the remainder when 73 × 75 × 78 × 57 × 197 × 37 is divided by 34.
[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.]
[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.]
Required remainder = 32
Solution:
Remainder,[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.]
[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.]
Required remainder = 32
2. Find the remainder when 6799 is divided by 7.
63 is divisible by 7 for any power, so required remainder will depend on the power of 4
Required remainder
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.
Solution:
63 is divisible by 7 for any power, so required remainder will depend on the power of 4
Required remainder
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.
3. Let N = 1421 × 1423 × 1425. What is the remainder when N is divided by 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,
= Remainder ⇒ -1 × -3 = 3 [Here, we have taken negative remainder.] So, required remainder will be 3.
Note: When, it gives positive remainder as 9 and it also give a negative remainder -3. As per our convenience,we can take any time positive or negative remainder.
Solution:
Remainder,[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,
= Remainder ⇒ -1 × -3 = 3 [Here, we have taken negative remainder.] So, required remainder will be 3.
Note: When, it gives positive remainder as 9 and it also give a negative remainder -3. As per our convenience,we can take any time positive or negative remainder.
4. Three numbers are in ratio 1 : 2 : 3 and HCF is 12. The numbers are:
Each one's common factor is HCF.
And here HCF = 12,
hence, the numbers are 12, 24 and 36.
Alternatively,Let the numbers be x, 2x and 3x.
The HCF in x, 2x and 3x is x because 1, 2, 3 are prime.
Hence,
x = 12; then the other numbers are 24 and 36.
Solution:
Since, the numbers are given in the form of ratio that means their common factors have been cancelled.Each one's common factor is HCF.
And here HCF = 12,
hence, the numbers are 12, 24 and 36.
Alternatively,Let the numbers be x, 2x and 3x.
The HCF in x, 2x and 3x is x because 1, 2, 3 are prime.
Hence,
x = 12; then the other numbers are 24 and 36.
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?
LCM of 12, 15, 18 and 20;
Hence, LCM =
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
=
= 900
Solution:
In this type of question, We need to find out the LCM of the given numbers.LCM of 12, 15, 18 and 20;
Hence, LCM =
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
=
= 900
6. Find the least number which will leaves remainder 5 when divided by 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
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
7. 76n- 66n, where n is an integer >0, is divisible by
Clearly, it is divisible by 127, 13 as well as 559
Solution:
Clearly, it is divisible by 127, 13 as well as 559
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?
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
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
10. Find the remainder when 496 is divided by 6.
Now, Remainder will depend only the powers of -2. So,
it is same as
it is same as
Now,
On dividing individually 16 we always get a remainder 4.
Hence, Required Remainder = 4
NOTE: When 4 has even number of powers, it will always give remainder 4 on dividing by 6.
Solution:
we can write it in this formNow, Remainder will depend only the powers of -2. So,
it is same as
it is same as
Now,
On dividing individually 16 we always get a remainder 4.
Hence, Required Remainder = 4
NOTE: When 4 has even number of powers, it will always give remainder 4 on dividing by 6.
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