Properties of The Number 137 - Mathematics Stack Exchange There are some other Properties for $137$ in Wikipedia I found another Properties such that $137=2^7+2^3+1$ or the only way to write the number $137$ as a summation of two square numbers is $137=4^2+11^2$ thanks for your advice and suggestions Edit: After reading comments and answers, I want to suggest a definition for such numbers like $137$
Dodecahedral number visualization - Mathematics Stack Exchange Visualization (based on Oscar's and Ed's answers) Tetrahedral and cubic numbers are much easier to grasp than dodecahedral numbers So, let’s build up 5 visualizations in parallel: Tetrahedral numbers Cubic numbers Dodecahedral numbers Dodecahedral' numbers (arranged halfway between a dodecahedron and tetrahedron) Dodecahedral'' numbers (arranged as a tetrahedron) Here are the 2D equivalents
Find an integer $r$ with $0 ≤ r ≤ 10$ such that $7^ {137 }≡ r (\text . . . Your working is fine You need to end by noting that $-5 \equiv 6 \pmod {11}$ since they asked for a residue between $0$ and $10$ An alternative approach would be: $7^ {137} \equiv (-4)^ {137} \equiv -2^ {274} \equiv - (2^ {5})^ {54} \cdot 2^4 \equiv - (32)^ {54} \cdot 16 \equiv - (-1)^ {54} \cdot 16 \equiv - 16 \equiv -5 \equiv 6 \pmod {11}$ I prefer this approach to reduce the base in