Question

2added by 1 minus 12445544added by 4765754765875435645

Answer 1

Answer:

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Step-by-step explanation:

What is the remainder when 123456...424344 is divided by 45?

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Let the given number be x.

Note that 1+2+...+43+44=(44*45)/2 = 990. This implies that the given number is divisible by 9 which means that if we add or subtract any multiple of 9 to it, it will remain divisible by 9. Consider x-9 (it'll be of the form 123456...424335). Clearly, this number is divisible by 9, and 5; and hence by 45 (since 5 and 9 are coprime).

Hence we may conclude that the given number gives a remainder of 9 when divided by 45.

Edit: This is to explain my claim as to why x is divisible by 9. The usual divisibility rule of 9 states that the remainder given by a number with 9 is equal to the remainder given by its sum of digits with 9. But calculating the sum of digits of x is not very easy to calculate. So, instead of calculating the remainder (1+2+3+...4+3+4+4) gives with 9, I've calculated the remainder (1+2+3+...+4*10+3+4*10+4) gives with 9. Note that the expressions in brackets will give equal remainders with 9, because 10 gives a remainder of 1 with 9.

(Thanks to for pointing this out).