Percentage division

Algebra Level 3

1 % ÷ 2 % ÷ 3 % ÷ ⋯ ÷ 98 % ÷ 99 % \large {1\%} \div{2\%} \div {3\%} \div \dots \div{98\%} \div {99\%}

If the expression above is equal to 10 0 a b ! \dfrac{100^{a}}{b!} , where a a and b b are integers, find the minimum value of a + b a+b .


The answer is 196.

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Nihar Mahajan by Nihar Mahajan , 20, India

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1 solution

The expression is equivalent to:

1 100 × 100 2 × 100 3 × 100 4 × . . . × 100 99 = 1 × 1 2 × 100 3 × 100 4 × . . . × 100 99 = 1 × 1 2 × 10 0 1 ( = 3 − 2 ) 3 × 100 4 × . . . × 100 99 = 1 × 1 2 × 10 0 2 ( = 4 − 2 ) 3 × 4 × 100 5 × . . . × 100 99 = 1 × 1 2 × 10 0 3 ( = 5 − 2 ) 3 × 4 × 5 × 100 6 × . . . × 100 99 = 10 0 99 − 2 99 ! = 10 0 97 99 ! \begin{aligned} \frac{1}{100} \times \frac{100}{2} \times \frac{100}{3} \times \frac{100}{4} \times ... \times \frac{100}{99} & = 1 \times \frac{1}{2} \times \frac{100}{3} \times \frac{100}{4} \times ... \times \frac{100}{99} \\ & = 1 \times \frac{1}{2} \times \frac{100^{\color{#D61F06}{1(=3-2)}}}{\color{#D61F06}{3}} \times \frac{100}{4} \times ... \times \frac{100}{99} \\ & = 1 \times \frac{1}{2} \times \frac{100^{\color{#D61F06}{2(=4-2)}}}{3\times \color{#D61F06}{4}} \times \frac{100}{5} \times ... \times \frac{100}{99} \\ & = 1 \times \frac{1}{2} \times \frac{100^{\color{#D61F06}{3(=5-2)}}}{3\times 4 \times \color{#D61F06}{5}} \times \frac{100}{6} \times ... \times \frac{100}{99} \\ & = \frac{100^{99-2}}{99!} \\ & = \frac{100^{97}}{99!} \end{aligned}

⇒ a + b = 97 + 99 = 196 \Rightarrow a + b = 97+99 = \boxed{196}

6 Helpful 1 Interesting 0 Brilliant 0 Confused

I think the answer like this. 1%:2%:3%:...:99% =1/100 x 100/2 x 100/3 x ... x 100/99 =100^98/100! a=98 b=100 a+b=100 + 98 =198

Ananda Ugracena - 5 years, 4 months ago

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I think like that. How about you guys?

Ananda Ugracena - 5 years, 4 months ago

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Check my solution again.

Chew-Seong Cheong - 5 years, 4 months ago

I think the answer should be 197 :/ but im not sure

Irvine Dwicahya - 5 years, 5 months ago

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Hey friend , why do you think that the answer should be 197? What are your values for a,b?

Nihar Mahajan - 5 years, 5 months ago

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I think they are mistaken and assumed the problem to be till 100 like other convectionl problems. I also got 197 as the answer in first attempt with a=98 but it actually a=97.

Chaitnya Shrivastava - 5 years, 5 months ago

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@Chaitnya Shrivastava – Check my solution again.

Chew-Seong Cheong - 5 years, 4 months ago

Check my solution again.

Chew-Seong Cheong - 5 years, 4 months ago

Check my solution again.

Chew-Seong Cheong - 5 years, 4 months ago

i aslo think answer must be 197

Harish Yadav - 5 years, 5 months ago

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Check my solution again.

Chew-Seong Cheong - 5 years, 4 months ago

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