HAPPY NEW YEAR!
Every year introduces us to a special number worth celebrating, and with it comes the chance to dive into its fascinating mathematical properties and expressions.
Let’s explore 2025’s captivating mathematical features and patterns.
A Perfect Square
2025 = 45²
The year 2025 is a perfect square, resulting from squaring 45. The last occurrence was in 1936, and the next one will be in 2116.
If we increment the digits of 45 and square this, something interesting happens: (45+10+1)2=452+1000+100+10+1, that is if we increment all the digits of 2025 by 1 we obtain a new square number 3136=562
Sum of the Digits is Also a Square
2+0+2+5=9
Any Two Consecutive Digits are Prime
2+0=2, 0+2=2 and 2+5=7
Remains Square When Most Significant Digit is Removed
025 is square
Sum of Cubes from 1 to 9
2025 = 1³ + 2³ + 3³ + ··· + 9³
This showcases the elegance of cubic numbers adding up to total 2025.
Square Which is Not the Sum of Two Primes
This is rarer than you may think!
2025=(20+25)2
Smallest Multiple of 15 with 15 Divisors
15×135
Divisible by all the factors of 15
Sum of the First 45 Odd Numbers
2025 = 1 + 3 + 5 + ··· + 89
The sum of the first 45 odd numbers equals 2025.
A Harshad Number
2 + 0 + 2 + 5 = 9, and 2025 ÷ 9 = 225
Since 2025 can be evenly divided by the sum of its digits (9), it is classified as a Harshad number.
A Triangular Square
45 = (9 × 10) ÷ 2
2025 is the square of 45, which also happens to be the 9th triangular number.
Sum of Two Squares
272+362=2025
Sum of Three Squares
2025 = 40² + 20² + 5²
2025 can be represented as the sum of three perfect squares.
Product of Two Squares
92×52=2025
Smallest Number with Exactly 15 Odd Divisors
1, 3, 9, 15, 81, 45, 729, 105, 225, 405, 59049, 315, 531441, 3645, 2025, 945, 43046721, 1575, 387420489, 2835, 18225, 295245, 31381059609, 3465, 50625, 2657205, 11025, 25515, 22876792454961, 14175, 205891132094649, 10395, 1476225, 215233605
Number of odd divisors is itself an odd divisor
1, 2, 4, 8, 9, 16, 18, 32, 36, 64, 72, 128, 144, 225, 256, 288, 441, 450, 512, 576, 625, 882, 900, 1024, 1089, 1152, 1250, 1521, 1764, 1800, 2025, 2048, 2178, 2304, 2500, 2601, 3042, 3249, 3528, 3600, 4050, 4096, 4356, 4608, 4761, 5000, 5202, 5625, 6084….
Product of Proper Divisors of its Square Root
45, i.e. 1×3×5×9×15
The Kaprekar Connection
2025 = 45² and 20 + 25 = 45
By splitting 2025 into 20 and 25, their sum equals 45, which relates to its Kaprekar property.
Power Sum Representation
2025 = 2¹⁰ + 2⁹ + 2⁸ + 2⁷ + 2⁶ + 2⁵ + 2⁴ − 2³ + 2² − 2¹ − 2⁰
A creative arrangement of powers of 2 with alternating additions and subtractions sums to 2025.
Inventive Arithmetic Representations
2025 = 12 × 3 + 45 × 6 × 7 + 89 + 10
2025 = 10 + (9 + 8 × 76 + 54) × 3 + 2 × 1
2025 = 12 × 3 + (4 + 5) × (6 + 7) × (8 + 9)
2025 = 9 × 8 + 76 + 54 × 3 + 2 × 1
Creative formulas show the versatility in expressing 2025.
Playful Mathematical Explorations
2025 = (1 + 1)¹¹ − 11 − 11 − 1
2025 = (2 × 2² + 2 ÷ 2)²
2025 = 3 × (3 × (3 + 3)³ + 33)
2025 = (44 + 4 ÷ 4) × (4 + 4) ÷ 4
2025 = 5 × (5 × (5 × 5 + 55) + 5)
2025 = 6 × 6 × 6 × 6 + (6 × 6 ÷ (6 + 6))⁶
2025 = (7 + 7 + 7 ÷ 7) × ((7 + 7)⁷ ÷ 7 + 7)
2025 = 88 × (8 + 8 + 8 − 8 ÷ 8) + 8 ÷ 8
2025 = 9 × ((9 + 9) × (9 + 9) − 99)
Composite
Can be expressed as the product of its’ prime factors. Its prime factorization is:
2025
2025=34×52
2025 = 3^4 x 5^22025=34×52
Deficient, Evil, Odd and Powerful
2025 is an Odd Refactorable Number
A “refactorable number” is a number n that is divisible by the count of it’s divisors.
For example, 18 is a refactorable number, because it’s divisors are: 1, 2, 3, 6, 9 and 18 (6 divisors) and 18/6 is an integer.
Lucas 9-step number
The Lucas 9-step sequence is defined as:
L(n)=L(n−1)+L(n−2)+L(n−3)+⋯+L(n−9),L(n) = L(n-1) + L(n-2) + L(n-3) + … + L(n-9),L(n)=L(n−1)+L(n−2)+L(n−3)+ … +L(n−9),
where the first 9 terms are usually initialized as:
L(1)=L(2)=…=L(9)=1
1, 3, 7, 15, 31, 63, 127, 255, 511, 1013, 2025
Magic Squares
A magic square is a square grid filled with distinct positive integers such that the sum of the numbers in each row, column, and diagonal is the same. This constant sum, 2025, is called the magic constant.
Here are some unique representations of fourth-order, fifth-order, sixth-order, ninth-order, tenth-order, and fifteenth-order magic squares whose magic sum is 2025.




These expressions showcase the playful side of math with intriguing ways to arrive at 2025. Explore these with your students to show the beautiful way numbers can work together!