QUESTIONS ON REMAINDERS & DIVISIBILITY (PART-II)

QUERY 12

25²⁵ is divided by 26, the remainder is?

A) 1
B) 2
C) 24
D) 25

RONNIE BANSAL
Use remainder theorem. Here 25²⁵ is the polynomial and the divisor is 26.
We can write 26 = 25+1 = 25 – ( -1)
So the remainder is ( -1)²⁵= -1. But we don’t take the remainder a negative term; so add it to the divisor.
So the remainder is 26 +( -1) = 25 (option ‘D’)

QUERY 13

When (67⁶⁷ + 67) is divided by 68, the remainder is?

A) 1
B) 63
C) 66
D) 67

RONNIE BANSAL
The given expression is in the form x⁶⁷ + x ……….(a polynomial in x)
Now 68 = 67 + 1; means x +1

So according to the remainder theorem when a polynomial is divided by another of the form          x + 1, the remainder is equal to p(­1) where p is the polynomial itself.
So the remainder is ­1⁶⁷ + (­1) = ­1 + (­1) = ­2

But the remainder should not be described negative of a number; in such a situation it is added to the divisor to find the actual.

So the remainder is ­2 + 68 = 66 (option ‘C’)

QUERY 14

What is the remainder when [(9¹⁹) + 6] is divided by 8

A) 6
B) 7
C) 0
D) 3

RONNIE BANSAL
The given expression is in the form (x¹⁹) + c; where ‘c’ is a constant ……….(a polynomial in x).

Now 8 = 9 ­ 1; means a polynomial in the form of x ­ 1

So according to the remainder theorem when a polynomial is divided by another of the form     x ­ 1, the remainder is equal to p(1) where p is the polynomial itself. So using remainder theorem, the remainder is (1¹⁹) + 6 = 1 + 6 = 7 (option ‘B’)

Also see it and compare:
When (67⁶⁷ + 67) is divided by 68, the remainder is?

A) 1
B) 63
C) 66
D) 67

RONNIE BANSAL
The given expression is in the form x⁶⁷ + x ……….(a polynomial in x)
Now 68 = 67 + 1; means x +1

So according to the remainder theorem when a polynomial is divided by another of the form          x + 1, the remainder is equal to p(­1) where p is the polynomial itself.
So the remainder is ­1⁶⁷ + (­1) = ­1 + (­1) = ­2

But the remainder should not be described negative of a number; in such a situation it is added to the divisor to find the actual.

So the remainder is ­2 + 68 = 66 (option ‘C’)

QUERY 15

Find the remainders in
1. 2¹¹/5

A) 0
B) 1
C) 2
D) 3

2. 7⁷/2⁴

A) 3
B) 5
C) 7
D) 1

RONNIE BANSAL
1. 2¹¹/5
In questions like this we should avoid using the remainder theorem as it can really be difficult when the power of a number (greater than 1) which is derived from the remainder theorem is so high. Better convert the base in powers of such numbers which are easily divisible by the divisor, like:

2¹¹/5 = 2⁴ x 2⁴ x 2³= 16 x 16 x 8

On dividing 16 by 5 we get 1 as the remainder; and if 8 is divided by 5 we get 3
So the multiplication of all the remainders
= 1 x 1 x 3 = 3 which is our answer (option ‘A’)

2. 7⁷/2⁴ = 7² x 7² x 7² x 7/16
= 49 x 49 x 49 x 7/16
Now the remainder on dividing 49 by 16 =1

The multiplication of all the remainders 1 x 1 x 1 x 7 = 7 (option ‘C’)

Let’s take another example:
Remainder when 2³¹  divided by 5?

MAHA GUPTA
2³¹ = [(2⁴)⁷]*2³
= (16⁷)*8

On dividing 16 by 5 we get 1 as the remainder, so the remainder when 16⁷ is divided by 5 = 1⁷ = 1; and if 8 is divided by 5 we get 3 as the remainder.

So the remainder needed = 1*3 = 3

NOTE: If the divisor is 2, 5 or 10, one needs to know only the unit digit of the dividend and divide it by the divisor to find the remainder. In this case 5 is the divisor, so just find the unit digit in 2³¹, which is 8 of course. So when 8 is divided by 5, the remainder is 3 (answer)

QUERY 16

What is remainder when the product of 177, 414 & 837 divided by 12

A) 9
B) 0
C) 6
D) 1

NIRMAL singh
In such a question we divide each number by the divisor and keep dividing the product of all remainders thus found by the same divisor till we get a number smaller than that divisor, that number is our answer.

So the remainders of 177, 414 & 837 when divided by 12 are 9, 6 & 9 respectively
Now their product = 9 x 6 x 9
When divided by 12 we can write it as 54 x 9/12
Again the remainders are 6 x 9
On division of it by 12 it will be 54/12

We see that the remainder is 6; and this is the answer. (option ‘C’)

QUERY 17

What is the remainder when [7^{(4n + 3)}]*6ⁿ is divided by 10; where ‘n’ is a positive integer.

A) 2
B) 4
C) 8
D) 6

JAYANTCHARAN CHARAN
[7^{(4n + 3)}]6ⁿ
= 7⁴ⁿ x 7³ x 6ⁿ
= 49²ⁿ x 7³ x 6ⁿ
= When each factor is divided by 10 the remainders in each case = (-1)²ⁿ, 3, and 6                                                   (6 when raised to the power of any natural number is divided by 10 always gives remainder as 6 itself)

So, all the remainders thus found above are 1, 3 and 6
So their multiplication= 1*3*6= 18
So the remainder after 18 has been divided by 10 = 8 (option ‘C’)

Harmeet Singh
The expression is [7^(4n + 3) ]* [6^n]

To find the remainder when any expression is divided by 10 is SIMILAR to just finding the last digit, i.e. unit digit of the expression only.

Now 7^(4n + 3) = (7^4n )* (7^3)
Thus When 7^4n divided by 10 for all ‘n’ will give unit digit 1; and When 7^3 divided by 10 will give units digit 3 as 7^3 = 343
6^n for all ‘n’ gives units digit 6

So, their product is 1*3*6 = 18
So the remainder after 18 has been divided by 10 = 8 (option ‘C)

QUERY 18

Which two digit number when divided by 3 gives remainder of 1; when divided by 4 gives remainder of 2; when divided by 5 gives remainder of 3; and when divided by 6 gives remainder of 4? [Type # 2, see query 8]

A) 60
B) 58
C) 56
D) 50

RONNIE BANSAL
One of the divisors is 5 giving remainder of 3; means the unit digit of that two digit number is either 3 or 8
But 3 is not possible here as one of the divisor is 4 giving remainder of 2; because the unit digit then must not be odd.
So the possible numbers are 18, 28, 38, 48, 58, 68, 78, 88 & 98

As 3 is a factor of 6; so we need not check the numbers with 3.
Therefore we need to check them considering 4 and 6 here. Now it can easily be seen that 58 is such number. (option ‘B’)

TRICK (see query 8)
When the difference between divisor and remainder is same like 3 – 1 = 4 – 2 = 5 – 3 = 6 – 4=2;
take LCM of divisors and subtract the common difference from it
Now the LCM of 3, 4, 5, 6 = 60

Therefore the required number here = 60 – 2 = 58 (option ‘B’)

QUERY 19

Numbers 11284 and 7655, when divided by a certain number of three digits, leave the same remainder. Find that number of 3 digits and their sum.

A) 191 & 11
B) 911 & 11
C) 181 & 10
D) 811 & 10

RONNIE BANSAL
One has to remember that each factor of the difference of two numbers gives the same remainder if those numbers are divided by it.

Now the difference here = 11284 – 7655 = 3629
Factors of 3629 are 1, 19, 191 and 3629
But we have to find the three digit number here, so 191 is the required number and their sum is 1+9+1 = 11 (option ‘A’)

QUERY 20

64329 is divided by a certain number. While dividing, the numbers 175, 114 and 213 appear as three successive remainders, the divisor is?

A) 184
B) 224
C) 234
D) 6250

RONNIE BANSAL
We have three remainders, means the number comprising of the first digits i.e. 643 was divided first and we got 175 as the remainder.

Now according to DIVIDEND = DIVISOR x QUOTIENT + REMAINDER
=> DIVISOR x QUOTIENT = DIVIDEND – REMAINDER
=> DIVISOR x QUOTIENT = 643 – 175 = 468

We see that 468 is divisible by 234 only among all the answer options; so 234 (option ‘C’) is the divisor we need.