Question

find the greatest number which will divide 410, 751, 1030 so as to leave the reminder 7 in each case

Answer 1

If all the given numbers leave 7 when divided by a particular number, then the numbers when subtracted by 7 divides leaving no remainder.
By implying the above, the numbers will be 403, 744, 1023 respectively.
The H.C.F. of these numbers, i.e., the greatest number which divides these numbers is '31'.
Therefore, the greatest number which will divide 410, 751, 1030 so as to leave the reminder 7 in each case is "31".

Answer 2

For finding no. dividing 410 , 751 , 1040 leaving remainder 7, the HCF of the no. will be (410-7), (751-7), (1030-7)

the h.c.f. of 403, 744, 1023 is 31
hence the no is 31