Question:The H.C.F of two numbers is 11 and their L.C.M is 693. If one of the numbers is 77, find the other. 

The other number = `((11xx693)/77) = 99`

Question:Find the largest number that can exactly divide `513, 713` and `1107`. 

Required number 
= H.C.F. of `513, 783` and `1107`.
Now, `513 = 3^3 xx 19,` 
` 783 = 3^3 xx 29,` and 
`1107 = 3^3 xx 41`
`:.` H.C.F. = `3^3 = 27`
Hence, the required number is `27`

Question:Find the greatest number which can divide 284, 698 and 1618 leaving the same remainder 8 in each case. 

Required number = 
H.C.F. of `(284 - 8), (698 - 8)` & `(1618 - 8)`
= H.C.F. of `276, 690` and `1610.`

Now, 
`276 = 2^2 xx 3 xx 23,`
`690 = 2 xx 3 xx 5 xx 23,` 
`1610 = 2 xx 5 xx 7 xx 23`

`:.` H.C.F. of `276, 690` & `1610` is `23`.
Hence, the required number is `23`.

Question:Find the largest number which divides 62, 132 and 237 to leave the same reminder in each case. 

Required number = H.C.F of (132-62), (237-132) & (237-62)
                       = H.C.F of 70, 105 & 175 = 35.

Question:Find the largest number of four digits  exactly divisible by 12, 15, 18 and 27. 

The largest number of four digits is 9999.
Required number must be divisible by 1 c.m of 12, 15, 18, 27 i.e. by 540.
On dividing 9999 by 540, we get 279 as remainder.
`:.` Required number = (9999-279) = 9720.

Question:Find the smallest number of five digits exactly divisible by 16, 24, 36 and 54. 

The smallest number of five digits is 10000.
Required number must be divisible by 1 c.m of 16, 24, 36, 54 i.e. 432.
On dividing 10000 by 432, we get 64 as remainder.
`:.` Required number = 10000+ (432-64) = 10368.