Find the pattern for these equations, and then make up one more of your own, using the same pattern.
12 � 42 = 21 � 24
13 � 62 = 31 � 26
13 � 62 = 31 � 26
| We observe: | 12 x 42 = | 21 | x | 24 |
| is 12 reversed |
is 42 reversed |
We try, say, 12 x 52 ≟ 21 x 25 → 624 ≠ 525. No good, we need to know more.
We observe: 12 x 42 = 21 x 24 and 13 x 62 = 31 x 26
We try 14 x 82 ≟ 41 x 28 → 1148 = 1148 Ah!
Algebraically:
| 1 | 2 | x | 4 | 2 | = | 2 | 1 | x | 2 | 4 |
| ↑ | ↑ | ↑ | ↑ | ↑ | ↑ | ↑ | ↑ | |||
| t | u | w | u | u | t | u | w |
→ (10t + u)(10w + u) = (10u + t)(10u + w)
→ 100tw + 10tu + 10uw + u2 = 100u2 + 10uw + 10tu + tw
→ 100tw + u2 = 100u2 + tw
→ 99tw = 99u2
→ tw = u2.
Try t = 8, w = 2, u = 4
84 x 24 ≟ 48 x 42 → 2016 = 2016.
Notice that 13 x 62 = 31 x 26 doesn’t quite follow this rule. What is the most general form of the rule?