Question

If the digits of a five-digit number in base N are increased by 1, the decimal value of the number increases by 16105. What is the sum of digits of N?

Answer 1

Answer:

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Answer 2

Given is the increased number in the decimal system for a five-digit number of base N, find the value of the sum of digits of N.

Explanation:

• Let the five digits of the number having base N be denoted by a, b, c, d, and e.
• Let the number in the decimal system be denoted by 'S'.
• Then the number in the decimal system can be given as,
•  $number->(abcde)_N \\->aN^4+bN^3+cN^2+dN+eN^0=(S)_{10}\\->aN^4+bN^3+cN^2+dN+e=S$    
• When each digit is increased by one we get,
• $number->((a+1)(b+1)(c+1)(d+1)(e+1))_N \\->(a+1)N^4+(b+1)N^3+(c+1)N^2+(d+1)N+(e+1)N^0=S+16105\\->aN^4+bN^3+cN^2+dN+e+ N^4+N^3+N^2+N+1=S+16105\\->N^4+N^3+N^2+N+1=16105 \\->N^4+N^3+N^2+N=16104\\->N(N^3+N^2+N+1)=11(1331+121+11+1)\\->N=11$  
• Hence we have $N=11$ and the sum of digits of N is $2$.