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1. One bus leaves every 12 minutes and another every 18 minutes. They leave together at 9:00. When do they next leave together?[3]
A 9:06B 9:30C 9:36D 9:54E 12:362. Which of these fractions lies between one third and one half?[3]
3. Red and blue counters are in the ratio 3 to 5. There are 40 counters altogether. How many are red?[4]
4. What is the units digit of 3 multiplied by itself 100 times, that is 3 to the power 100?[4]
A 1B 3C 7D 9E it cannot be found without a calculator5. What is the units digit of 7 multiplied by itself 2026 times, that is 7 to the power 2026?[5]
A it cannot be found without a calculatorB 1C 3D 7E 9Kangaroo Atlas · https://kangaroo-atlas.vercel.app · seed 1 · CC BY-NC-SA 4.0 · Independent project, not affiliated with Math Kangaroo in USA, NFP.
1. C — 9:36 KA-0041 Factors, multiples, LCM and GCD by listing
A I found the greatest common divisor of 12 and 18 instead of the least common multiple.B I added the two intervals together rather than looking for a time that is a multiple of both.D I found a common multiple but not the least one, landing on a later coincidence.E I multiplied the two intervals together instead of finding their least common multiple.2. C — 2/5 KA-0141 Number lines and betweenness
A I compared the bottom numbers and assumed a bigger denominator means a bigger fraction.B I saw that 4 lies between 3 and 2 and assumed one quarter therefore lies between the two fractions.D I checked that it is bigger than one third but never checked it against one half.E I compared the top numbers only and ignored what the bottoms do to the size.3. D — 15 KA-0024 Ratio as parts of a whole
A I found what one part is worth and gave that as my answer instead of multiplying it back up.B I divided 40 by the five blue parts instead of by the eight parts that make the whole.C I read the ratio 3 to 5 as meaning 3 out of every 5 and worked from that.E I found the value of one part correctly but then multiplied by 5 instead of 3, giving the blue counters.4. A — 1 KA-0149 Last-digit behavior of products and powers
B I assumed every power of 3 ends in 3.C I found the cycle 3, 9, 7, 1 but counted its positions from zero, landing one step early.D I used a remainder of 2 instead of 0 when dividing 100 by the cycle length of 4.E I assumed a number this large has no reachable last digit, rather than looking for a repeat.5. E — 9 KA-0037 Last-digit behavior of products and powers
A I assumed a number this large has no reachable last digit and gave up on the cycle.B I found the cycle 7, 9, 3, 1 but used a remainder of 0 when the remainder is actually 2.C I found the cycle but counted its positions starting at zero, shifting my answer along by one.D I assumed the units digit of any power of 7 is always 7.