What are the last two digits of 63*35*37*82*71*41*53?
Choices:
A) 10 B) 30 C) 40 D) 70 E) 80

Rationale: we know that to find the last two digits , we need to find the remainder we get when we divide the product by 100 . remainder of ( 63 * 35 * 37 * 82 * 71 * 41 ) / 100 note that we can simplify this expression by canceling out the 5 and 2 in the numerator and denominator . but before we do that , here is an important note : note : we can not just cancel off the common terms in the numerator and denominator to get the remainder . but , if we want to cancel off to simplify the question , we can do it , provided we remember to multiply it back again . so say , we want to find the remainder when 14 is divided by 10 i . e . 14 / 10 ( remainder 4 ) . but we cancel off the common 2 to get 7 / 5 . the remainder here will be 2 which is not the same as the remainder obtained by dividing 14 by 10 . but if we multiply 2 back by 2 ( the number we canceled off ) , the remainder will become 2 * 2 = 4 which is correct . take another example to reinforce this – what is the remainder when 85 is divided by 20 ? it is 5 . we might rephrase it as – what is the remainder when 17 is divided by 4 ( cancel off 5 from the numerator and the denominator ) . the remainder in this case is 1 . we multiply the 5 back to 1 to get the remainder as 5 which is correct . so keeping this very important point in mind , let ’ s go ahead and cancel the common 5 and 2 . we need the remainder of ( 63 * 7 * 37 * 41 * 71 * 41 * 5 * 2 ) / 10 * 5 * 2 remainder of ( 63 * 7 * 37 * 41 * 71 * 41 ) / 10 now using concept 2 , let ’ s write the numbers in form of multiples of 10 remainder of ( 60 + 3 ) * 7 * ( 30 + 7 ) * ( 40 + 1 ) * ( 70 + 1 ) * ( 40 + 1 ) / 10 remainder of 3 * 7 * 7 * 1 * 1 * 1 / 10 remainder of 147 / 10 = 7 now remember , we had canceled off 10 so to get the actual remainder so we need to multiply by 10 : 7 * 10 = 70 . when 63 * 35 * 37 * 82 * 71 * 41 is divided by 100 , the remainder is 70 . so the last two digits of 63 * 35 * 37 * 82 * 71 * 41 must be 80 . answer ( e )
Answer: E