Question
The number of lollipops in a box is between 60 and 100. If they are put into packets of 3, there
will be 1 lollipop left. If they are
put into packets of 5, there will be 1 lollipop left. If they are put into packets of 7, there
will be no lollipop left. How many
lollipops are there in the box?

Introduction
This
question seems to involve writing lists of numbers and finding the elusive
common number. Some school teacher did
just that starting from 3, 4, 5, 6 and 7, when the question already mentions
that the number is between 60 and 100.
To add insult to injury, the boy who got
this wrong in a test copied this teacher’s “model solution” as corrections! Sigh!
Is there a simple way to solve this mathematics
problem without rummaging through the entire haystack, as it were? The answer is: thankfully yes!
Solution
Solution
Ans: 91
lollipops
Explanation
First, I draw a table to analyse the remainders when divided by 3, 5 and 7. [using heuristics H02 and H03]. The question asks for a number with remainders (1, 1, 0). However, we can SHIFT the problem. [H09] If we consider one less than the required number, the remainders are (0, 0, 6). We look for a number with such a profile.
Having zero remainders under division by 3 and by 5, this number must be a multiple of 15. Trying 15 [H07], I get (0, 0, 1) because 15 = 2´14+1. To get a remainderprofile of (0, 0, 6), I multiply by 6, and I get 15´6 = 90. Now I just SHIFT back 1 to get the required answer!
List of Heuristics Used
Explanation
First, I draw a table to analyse the remainders when divided by 3, 5 and 7. [using heuristics H02 and H03]. The question asks for a number with remainders (1, 1, 0). However, we can SHIFT the problem. [H09] If we consider one less than the required number, the remainders are (0, 0, 6). We look for a number with such a profile.
Having zero remainders under division by 3 and by 5, this number must be a multiple of 15. Trying 15 [H07], I get (0, 0, 1) because 15 = 2´14+1. To get a remainderprofile of (0, 0, 6), I multiply by 6, and I get 15´6 = 90. Now I just SHIFT back 1 to get the required answer!
List of Heuristics Used
H02. Use a
diagram / model
H03. Make a
systematic list
H07. Use guess
and check
H09. Restate the problem in another way
Related problem here.
For Your Information
In solving this question, I did not really use any Chinese Remainder Theorem or advanced mathematics. I merely used heuristics that can be understood by most people, including the parents helping them and the teachers marking the test scripts. J
In solving this question, I did not really use any Chinese Remainder Theorem or advanced mathematics. I merely used heuristics that can be understood by most people, including the parents helping them and the teachers marking the test scripts. J
The kids? Oh! They will be fine. They will learn well if we equip them with powerful thinking skills but do not interfere too much. Shift happens! J
Related problem here.
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