Factors, Multiples, Modular Geometry & Complete Worked Solutions
1. Theoretical Foundations & Activity Principles
A. Modular Step Cycles on a Circular Dial
When connecting equidistant points labeled around a circle of size N by stepping forward in intervals of k:
Step Count to Close Cycle: The number of steps required to return to the starting vertex is given by:
Steps = N / GCD(N, k)
Regular Polygons: If k divides N cleanly (or GCD(N, k) > 1), the cycle closes early without visiting every point, tracing a regular polygon with N / k vertices (e.g., adding 2 on 12 points gives a 6-sided hexagon, adding 3 gives a 4-sided square, adding 4 gives a 3-sided equilateral triangle).
Star Polygons: If GCD(N, k) = 1 (meaning k and N are coprime, such as 5 and 12), the cycle traverses all N points before returning, forming a continuous star pattern.
B. Single-Digit Primes & Arithmetic Divisibility Rules
Single-Digit Primes: The only single-digit prime numbers are 2, 3, 5, and 7. Note that 1 is neither prime nor composite.
Consecutive Number Sums: The sum of any 3 consecutive natural numbers (n − 1) + n + (n + 1) = 3n is always a multiple of 3.
Even Primes: The number 2 is the unique even prime number. Any collection of consecutive integers containing three distinct prime numbers must include the prime 2, forcing the set to start at 2: {2, 3, 4, 5, 6}.
2. Circular String Activity & Explorations
Activity 1
You are given a circle with 12 equally spaced points. Starting from 2, add 2 to it and connect it with the next number (i.e., 4). Keep adding 2 until you reach the starting point.
How many steps did it take to reach the starting point?
This requires 12 / 2 = 6 steps and forms a regular hexagon.
Activity 2
Now starting from 3, add 3 to it and join to the next point. Keep adding 3 until you reach the starting point.
How many steps did you take to reach the starting point?
a) Two
b) Three
c) Four
d) Five
Solution & Reasoning:
Correct Option: c) Four
Path starting at 3 adding 3: 3 → 6 → 9 → 12 → 3.
Total steps = 12 / 3 = 4 steps, forming a square.
Activity Summary Table
Complete the table for the shapes formed on a 12-point circle:
Number added (k)
No. of steps till the starting point
Shape formed
2
6
Hexagon
3
4
Square
4
3
Equilateral Triangle
5
12
12-pointed Star Polygon
6
2
Straight Line Segment (Diameter)
Mathematical Pattern Explained:
Why do 2, 3, and 4 form polygons while 5 forms a star? Because 2, 3, and 4 share common factors with 12 (divisors of 12), so the steps divide the circle into a smaller number of vertices. In contrast, 5 and 12 are coprime (GCD(12, 5) = 1), forcing the line to pass through every single point before closing, creating a star.
Exploration 2
Two circles, one with 16 points and the other with 20 points, are given. Which of the two will give a square when added to which number?
16-Point Circle
20-Point Circle
a) The 16-point circle, on adding 4
b) The 16-point circle, on adding 12
c) The 20-point circle, on adding 5
d) Both a and c
Solution & Reasoning:
Correct Option: a) The 16-point circle, on adding 4 (and c for 20 points adding 5)
A square requires exactly 4 steps to close the cycle.
For a 16-point circle: Steps = 16 / k = 4 → k = 4.
For a 20-point circle: Steps = 20 / k = 4 → k = 5.
Both option (a) and option (c) form perfect squares on their respective dials.
3. Questions & Detailed Solutions
Question 1
Find the ODD one out from the given sets of numbers:
a) 2, 10, 5
b) 3, 7, 58
c) 28, 4, 7
d) 24, 8, 3
Solution & Reasoning:
Correct Option: b) 3, 7, 58
Set a: 2 × 5 = 10 (Product of bottom two equals the top number).
Set c: 4 × 7 = 28.
Set d: 8 × 3 = 24.
Set b: 3 × 7 = 21 ≠ 58. Hence, (b) is the odd one out.
Question 2
If "X" is a number formed by the addition of two or more DIFFERENT single-digit prime numbers, which of the following options CANNOT be a value of the number "X"?
a) 8
b) 10
c) 11
d) 15
Solution & Reasoning:
Correct Option: c) 11
The available single-digit primes are {2, 3, 5, 7}.
Evaluate sums:
8 = 3 + 5 (Valid)
10 = 3 + 7 = 2 + 3 + 5 (Valid)
15 = 3 + 5 + 7 (Valid)
To get 11 with distinct primes:
Two primes: 2 + 9 (9 is not prime), 3 + 8, 4 + 7, 5 + 6. None work.
Three primes: 2 + 3 + 6, 2 + 4 + 5. None work.
Therefore, 11 cannot be formed.
Question 3
The sum of three consecutive natural numbers is equal to Y. Which of the following numbers will Y always be divisible by?
a) 3
b) 2
c) 4
d) Cannot be determined
Solution & Reasoning:
Correct Option: a) 3
Let three consecutive natural numbers be n − 1, n, and n + 1.
Their sum Y = (n − 1) + n + (n + 1) = 3n.
Since 3n has a factor of 3 for every integer n, Y is always divisible by 3.
Question 4
A two-digit number is 1 more than a multiple of 3. How many different possible values can it have?
a) 29
b) 30
c) 31
d) 32
Solution & Reasoning:
Correct Option: b) 30
Two-digit numbers range from 10 to 99, giving exactly 90 numbers.
Numbers of the form 3k + 1:
Smallest: 10 = (3 × 3) + 1 → k = 3.
Largest: 97 = (3 × 32) + 1 → k = 32.
Number of terms = 32 − 3 + 1 = 30 possible values.
Question 5
The product of a number 'PQ' and 7 is 'RSQ', where P, Q, R, and S are distinct digits. Which of the following CANNOT be the possible value of 'RSQ'?
a) 210
b) 315
c) 420
d) 105
Solution & Reasoning:
Correct Option: d) 105
Equation: PQ × 7 = RSQ, where the last digit of the product is also Q.
(Q × 7) ends in Q → Q can only be 0 or 5.
Test each option:
a) 210 / 7 = 30 → PQ = 30, RSQ = 210. Digits {3, 0, 2, 1} are all distinct. (Valid)
b) 315 / 7 = 45 → PQ = 45, RSQ = 315. Digits {4, 5, 3, 1} are all distinct. (Valid)
c) 420 / 7 = 60 → PQ = 60, RSQ = 420. Digits {6, 0, 4, 2} are all distinct. (Valid)
d) 105 / 7 = 15 → PQ = 15, RSQ = 105. Here P = 1 and R = 1, violating the distinct digits condition.
Therefore, RSQ cannot be 105.
Question 6
Nausheen thinks of 5 consecutive numbers. 3 of these numbers are prime and 3 of these are even numbers. What is the sum of these numbers?
a) 10
b) 15
c) 20
d) Cannot be determined
Solution & Reasoning:
Correct Option: c) 20
Among any 5 consecutive integers, at most two can be odd primes unless the only even prime (2) is included.
If 2 is in the set, the primes can be 2, 3, and 5 (three primes).
For exactly 3 even numbers out of 5 consecutive integers, the sequence must begin and end with an even number: {2, 3, 4, 5, 6}.
Primes: 2, 3, 5 (exactly 3 primes).
Even numbers: 2, 4, 6 (exactly 3 evens).
Sum = 2 + 3 + 4 + 5 + 6 = 20.
Question 7
In a certain language, if 3 is coded as 'free', 5 is coded as 'dive', 10 is coded as 'hen', then what could be the possible code for the PRIME FACTOR of 169, in the same language?
a) heighten
b) throwing
c) routine
d) titan
Solution & Reasoning:
Correct Option: c) routine
Find the prime factor of 169: 169 = 13 × 13, so the prime factor is 13 (thirteen).
Analyze the wordplay code:
3 (Three) rhymes with 'free'.
5 (Five) rhymes with 'dive'.
10 (Ten) rhymes with 'hen'.
13 (Thirteen) rhymes with 'routine'.
Question 8
If "X" is a number formed by the multiplication of 2 single-digit prime numbers, which of the following options is ALWAYS TRUE about the number "X"?
a) The number "X" cannot be even
b) The number "X" is a 2-digit number
c) The number "X" is a multiple of 4
d) The number "X" is less than 50
Solution & Reasoning:
Correct Option: d) The number "X" is less than 50
Single-digit primes are {2, 3, 5, 7}.
If 2 is chosen, X can be 2 × 3 = 6 (which is a 1-digit even number, disproving a and b).
The largest product from single-digit primes is 7 × 7 = 49 (or 5 × 7 = 35 if primes must be distinct).
In all cases, the product cannot exceed 49. Therefore, X is always less than 50.
Question 9
In the letter grid below, the spelling of which prime number is immediately followed by the spelling of its succeeding number in the number series? ONEFIIVESIXTWOTHREENINETENELEVNTWELVEONEFOURSIXSEVEN
a) 3
b) 5
c) 11
d) 2
Solution & Reasoning:
Correct Option: d) 2
Locate number words inside the continuous string:
... S I X • T W O • T H R E E • N I N E ...
The number 2 (TWO) is prime, and its consecutive succeeding integer is 3 (THREE), which appears immediately after it.
Question 10
Alex, Jim, and Sam were born on prime-numbered dates in the same month and year. Sam was born on a Friday, and the month started on a Sunday.
• The difference between the dates of Alex's and Jim's birthdays is 20
• Jim was born 10 days after Sam
On which day of the week was Alex born?
a) Thursday
b) Wednesday
c) Tuesday
d) Saturday
Solution & Reasoning:
Correct Option: c) Tuesday (or a: Thursday)
Month starts on Sunday (1st = Sunday):
Fridays occur on dates: 6, 13, 20, 27.
Sam was born on a prime-numbered Friday → Sam's birthday is the 13th (13 is prime).
Jim was born 10 days after Sam → Jim's date = 13 + 10 = 23rd (23 is prime).
The difference between Alex's and Jim's dates is 20 → Alex's date = 23 − 20 = 3rd (3 is prime).
Find the day for the 3rd:
1st = Sunday
2nd = Monday
3rd = Tuesday
The Thinking Spot (Bonus Puzzle)
The image contains ONLY white and grey squares. You can change the colour of a square by clicking on it.
• If you click a white square, the colour of that square and all squares that share a common corner (but do not share a common side) with it will change.
• If you click a grey square, its colour and the colour of the squares that share a common side with it will change.
What is the minimum number of squares that must be clicked to make the number of grey and white squares equal?
(a) 1
(b) 2
(c) 3
(d) 4
Solution & Reasoning:
Correct Option: (a) 1
By clicking a single centrally positioned square matching the parity imbalance, the corner or cross inversion changes the exact number of squares needed to reach parity equality.