A number theory problem by أحمد الحلاق

Number Theory Level pending

Find the sum of digits of the number $2^{200} \times 3^4 \times 4^{55} \times 5^{312}$ .

9 7 1 3 5

1 solution

Zee Ell
Sep 28, 2016

$2^{200} × 3^4 × 4^{55} × 5^{312} = 2^{200} × 3^4 × 2^{110} × 5^{312} =$

$= (3^{4} × 5^2)× (2^{310} × 5^{310}) = (81 × 25) × 10^{310} = 2025 × 10^{310}$

Hence, the sum of the digits:

$2+0+2+5+0×310 = \boxed {9}$

I notice that the expression is a multiple of 9: $3^2$ is a factor

Hence the sum of the digits must be a multiple of 9. Since 9 is the only option out of the answers the answer must be 9. This is not a proper solution.

Jihoon Kang - 4 years, 8 months ago

While your (opportunistic) solution is shorter with these options, it wouldn't work if there was another option, divisible by 9 (say, 18 instead of the not so plausible 1). Neither would it work, if there were no options given (many (most?) questions on Brilliant are not in MCQ format).

Since I determine the exact number (written as 2025 followed by 310 zeros) and the sum of the digits from that, my solution is proper.

Zee Ell - 4 years, 8 months ago

A number theory problem by أحمد الحلاق

1 solution

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