1. **Problem statement:** Josh starts with a whole number (not 8 or 11) and counts by a whole number (not 5). Some of his numbers are 11, 32, and 46, but 24 is not one of his numbers. We need to find Josh's starting number. 2. **Understanding the problem:** Josh's numbers form an arithmetic sequence: $$a, a+d, a+2d, a+3d, \ldots$$ where $a$ is the starting number and $d$ is the step size. 3. **Given:** - $a \neq 8$ and $a \neq 11$ - $d \neq 5$ - The sequence contains 11, 32, and 46 - The sequence does not contain 24 4. **Use the arithmetic sequence formula:** $$a + nd = \text{terms in the sequence}$$ 5. **Set up equations for known terms:** Let $n_1, n_2, n_3$ be integers such that: $$a + n_1 d = 11$$ $$a + n_2 d = 32$$ $$a + n_3 d = 46$$ 6. **Find differences:** $$32 - 11 = 21 = (n_2 - n_1)d$$ $$46 - 32 = 14 = (n_3 - n_2)d$$ 7. **Since $d$ divides both 21 and 14, $d$ must be a common divisor of 21 and 14 other than 5.** 8. **Common divisors of 21 and 14:** 1, 7, and their negatives. Since $d \neq 5$, try $d=7$. 9. **Check $d=7$:** $$n_2 - n_1 = \frac{21}{7} = 3$$ $$n_3 - n_2 = \frac{14}{7} = 2$$ 10. **Find $a$ using $a + n_1 d = 11$:** Try $n_1=0$ (starting point at 11): $a=11$ (not allowed). Try $n_1=1$: $a = 11 - 7 = 4$ (allowed). 11. **Check if 24 is in the sequence starting at 4 with step 7:** $$4 + kd = 24 \Rightarrow 4 + 7k = 24 \Rightarrow 7k = 20 \Rightarrow k = \frac{20}{7}$$ not an integer, so 24 is not in the sequence. 12. **Conclusion:** Josh starts at 4 and counts by 7. **Final answer:** Josh starts with **4**.