Let N be the greatest number that will divide 1305, 4665 and 6905, leaving the same remainder in each case. Then sum of the digits in N is:

1. Let N be the greatest number that will divide 1305, 4665 and 6905, leaving the same remainder in each case. Then sum of the digits in N is:

A. 4 
B. 5
C. 6 
D. 8

Answer: Option A

Explanation:

N = H.C.F. of (4665 - 1305), (6905 - 4665) and (6905 - 1305)

= H.C.F. of 3360, 2240 and 5600 = 1120.

Sum of digits in N = ( 1 + 1 + 2 + 0 ) = 4

2. The greatest number of four digits which is divisible by 15, 25, 40 and 75 is:

A. 9000 
B. 9400
C. 9600 
D. 9800

Answer: Option C

Explanation:

Greatest number of 4-digits is 9999.

L.C.M. of 15, 25, 40 and 75 is 600.

On dividing 9999 by 600, the remainder is 399.

Required number (9999 - 399) = 9600.

3. The product of two numbers is 4107. If the H.C.F. of these numbers is 37, then the greater number is:

A. 101 
B. 107
C. 111 
D. 185

Answer: Option C

Explanation:

Let the numbers be 37a and 37b.

Then, 37a x 37b = 4107

ab = 3.

Now, co-primes with product 3 are (1, 3).

So, the required numbers are (37 x 1, 37 x 3) i.e., (37, 111).

Greater number = 111.

4. Find the HCF of 2⁴ X 5² X 7³ X 11², 7⁶ X 5⁴ X 2⁷ X 13, 5⁶ X 2² X 7⁴ X 17.

For this question we need to take common factors first, and then lowest powers.

Here common factors are 2, 5, 7.

Then write their lowest powers.

Answer = 2² x 5² x 7³.

5. Find the LCM of 2⁴ X 5² X 7³ X 11², 7⁶ X 5⁴ X 2⁷ X 13, 5⁶ X 2² X 7⁴ X 17.

For this question we need to write all the given numbers , and then write their Highest powers.

All the numbers are 2, 5, 7, 11, 13, 17.

Then write their Highest powers.

Answer = 2⁷ x 5⁶ x 7⁶ x 11² x 13 x 17.