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Question

How many four digit numbers are there such that when they are divided by 101 they have 99 as remainder? 


Solution

We observe that 1008 is the least number of 4 digits which gives a remainder 99 when divided by 101 and 9997 is the greatest.

So the numbers are 1008, 1109, 1210, 1311,……………..9997. These are in Arithmetic sequence in which first term a = 1008, common difference

d = 101 and the last term l= 9997. We use the formula l = a + (n-1)d to find n.

So, 9997 = 1008 + (n-1) 101 or 9997 - 1008 = (n-1) 101 or 8989 = (n-1)101. So, n-1 = 8989/101 = 89. This gives n = 90.

Therefore, there are 90 four-digit numbers which leave remainder 99 when divided by 101.

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