A Puzzle A Day: 2026-03-26

Dr. Arithmos is conducting a peculiar experiment using a series of numbered 'resonance tubes'. These tubes are numbered sequentially starting from 1. For a tube to be activated, its number must satisfy two unique properties when considered as a *three-digit number* (e.g., tube 7 is 007, tube 42 is 042, tube 123 is 123): 1. The sum of its digits must be perfectly divisible by 3. 2. The product of its digits must be perfectly divisible by 5. Dr. Arithmos needs to identify the 9th tube that will be activated in this experiment. Which tube number is it?
Correct: 102
To solve this, we need to systematically check tube numbers, considering each as a three-digit number (padding with leading zeros for single and double-digit numbers, e.g., 15 becomes 015, 7 becomes 007). We apply both conditions: **Condition 1:** The sum of its digits (S.D.) must be divisible by 3. **Condition 2:** The product of its digits (P.D.) must be divisible by 5. This is a crucial detail, as it implies that at least one of the three digits (hundreds, tens, or units place, including leading zeros) *must* be either 0 or 5. Let's list the activated tubes in order: 1. **Tube 15 (as 015):** S.D. = 0+1+5 = 6 (divisible by 3). P.D. = 0*1*5 = 0 (divisible by 5). **Activated.** 2. **Tube 30 (as 030):** S.D. = 0+3+0 = 3 (divisible by 3). P.D. = 0*3*0 = 0 (divisible by 5). **Activated.** 3. **Tube 45 (as 045):** S.D. = 0+4+5 = 9 (divisible by 3). P.D. = 0*4*5 = 0 (divisible by 5). **Activated.** 4. **Tube 51 (as 051):** S.D. = 0+5+1 = 6 (divisible by 3). P.D. = 0*5*1 = 0 (divisible by 5). **Activated.** 5. **Tube 54 (as 054):** S.D. = 0+5+4 = 9 (divisible by 3). P.D. = 0*5*4 = 0 (divisible by 5). **Activated.** 6. **Tube 60 (as 060):** S.D. = 0+6+0 = 6 (divisible by 3). P.D. = 0*6*0 = 0 (divisible by 5). **Activated.** 7. **Tube 75 (as 075):** S.D. = 0+7+5 = 12 (divisible by 3). P.D. = 0*7*5 = 0 (divisible by 5). **Activated.** 8. **Tube 90 (as 090):** S.D. = 0+9+0 = 9 (divisible by 3). P.D. = 0*9*0 = 0 (divisible by 5). **Activated.** 9. **Tube 102 (as 102):** S.D. = 1+0+2 = 3 (divisible by 3). P.D. = 1*0*2 = 0 (divisible by 5). **Activated.** Thus, the 9th tube to be activated is 102. Let's check the other choices: * **Tube 100 (as 100):** S.D. = 1+0+0 = 1 (NOT divisible by 3). Fails. * **Tube 125 (as 125):** S.D. = 1+2+5 = 8 (NOT divisible by 3). Fails. * **Tube 150 (as 150):** S.D. = 1+5+0 = 6 (divisible by 3). P.D. = 1*5*0 = 0 (divisible by 5). This tube is activated, but it is the 14th in the sequence (10th is 105, 11th is 108, 12th is 120, 13th is 135, 14th is 150), not the 9th.
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