EXAMPLE PROBLEMS 1

Problems:
1. If a number when divided by 296 gives a remainder 75, find the remainder when
37 divides the same number.
Method:
Let the number be ‘x’, say
∴x = 296k + 75, where ‘k’ is quotient when ‘x’ is divided by ‘296’
= 37 × 8k + 37 × 2 + 1
= 37(8k + 2) + 1
Hence, the remainder is ‘1’ when the number ‘x’ is divided by 37.

2. If (2^32)+1 is divisible by 641, find another number which is also divisible by ‘641’.
Method:                                           NOTE:^-POWER OF NUMBER
Consider 2^96+1 = (2^32)^3 + 1^3

= (2^32 +1)(2^64-2^32 +1)

From the above equation, we find that 296+1 is also exactly divisible by 641,
since it is already given that 232+1 is exactly divisible by ‘641’.


3.3. If m and n are two whole numbers and if m^n = 25. Find n^m, given that n ≠ 1

m^n = 25 = 5^2
∴m = 5, n = 2
∴n^m = 2^5 = 32

4.A number when successively divided by 9, 11 and 13 leaves remainders 8, 9 and
8 respectively.
Method:
The least number that satisfies the condition= 8 + (9×9) + (8×9×11) = 8 + 81 +
792 = 881

5.A number when divided by 19, gives the quotient 19 and remainder 9. Find the
number.
Let the number be ‘x’ say.
x = 19 × 19 + 9
= 361 + 9 = 370

6.Four prime numbers are given in ascending order of their magnitudes, the
product of the first three is 385 and that of the last three is 1001. Find the
largest of the given prime numbers.
The product of the first three prime numbers = 385
The product of the last three prime numbers = 1001
In the above products, the second and the third prime numbers occur in
common. ∴ The product of the second and third prime numbers = HCF of the
given products

HCF of 385 and 1001 = 77

∴Largest of the given primes =1001/77= 13

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