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2022 SMO Junior Answers

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Are there any formulas we should remember?

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Are there any formulas we should remember?

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Note that the difference of the two denominator numbers is the key first step.

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Worksheet 5. Reasoning behind summation of fractions

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Worksheet 4, Example 2. Two approaches to solve.

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For students who requested...
Note that once the length of PH is calculated as 10/3, the required area FDHP can be calculated by using the area of trapezium, where h = 3, the two parallel sides are 10/3 and 2 units.

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Attached are the 2 methods.

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Introduction to how (mod)7 is used to determine a remainder when 3²⁰⁰¹ is divided by 7

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Introduction to how (mod)7 is used to determine a remainder when 3²⁰⁰¹ is divided by 7

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For students interested to see how the (mod) method is used to determine a remainder. Note this is typically recommended knowledge for 16 to 18 years plus students,  but not taught as part of the Singapore Math syllabus.

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What is the remainder when 3 ²⁰⁰¹ is divided by 7?

Zi Heng Lim and Hagar El Bishlawi found a pattern when they raised 3 to a power, divided it by seven and found the remainder.

When you divide the exponent 2001 by the number 6, the answer will be 333 R3. The third number in the remainder pattern is 6, therefore 3 ²⁰⁰¹ divided by 7 has a remainder of 6, assuming that the pattern keeps repeating itself every six powers.

Can you prove that this pattern will keep repeating itself?

Pierce Geoghegan and Etienne Chan solved this problem using modulus arithmetic. This is Pierce's solution:

Initially we use the fact that 3 (mod 7) = 3 x [3 (mod 7)]

so

Now we have 3 ⁶ = 1 mod 7 and it follows that 3 ¹⁹⁹⁸ = (3 ⁶ ) ³³³ = 1 ³³³ (mod 7) = 1

But 3 ²⁰⁰¹ = 3 ¹⁹⁹⁸ x 3 ³ = 1 x 3 ³ (mod 7) = 6
therefore the remainder is 6 when 3 ²⁰⁰¹ is divided by 7.

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What is the remainder when 3 2001 is divided by 7? Zi Heng Lim and Hagar El Bishlawi found a pattern when they raised 3 to a power, divided it by seven and found the remainder. When you divide the exponent 2001 by the number 6, the answer will be 333 R3.