Answer :
Answer:
Step-by-step explanation:
I will find the squares of the given numbers using the diagonal method and then verify the results using the prime factorization method.
(i) 89
Diagonal method:
8 9
8 64 72
9 72 81
The diagonal sum is 64 + 81 = 145. The square of 89 is 89^2 = 7921.
Prime factorization method:
7921 = 89 × 89
(ii) 293
Diagonal method:
2 9 3
2 4 18 6
9 18 81 27
3 6 27 9
The diagonal sum is 4 + 81 + 9 = 94. The square of 293 is 293^2 = 85849.
Prime factorization method:
85849 = 293 × 293
(iii) 367
Diagonal method:
3 6 7
3 9 54 63
6 54 324 42
7 63 42 49
The diagonal sum is 9 + 324 + 49 = 382. The square of 367 is 367^2 = 134689.
Prime factorization method:
134689 = 367 × 367
(iv) 267
Diagonal method:
2 6 7
2 4 24 14
6 24 144 84
7 14 84 49
The diagonal sum is 4 + 144 + 49 = 197. The square of 267 is 267^2 = 71289.
Prime factorization method:
71289 = 267 × 267
(v) 109
Diagonal method:
1 0 9
1 1 0 9
0 0 0 0
9 9 0 81
The diagonal sum is 1 + 0 + 81 = 82. The square of 109 is 109^2 = 11881.
Prime factorization method:
11881 = 109 × 109
(vi) 247
Diagonal method:
2 4 7
2 4 16 28
4 16 64 112
7 28 112 196
The diagonal sum is 4 + 64 + 196 = 264. The square of 247 is 247^2 = 61009.
Prime factorization method:
61009 = 247 × 247
In all cases, the diagonal method and the prime factorization method produce the same results.
