1. **Problem statement:** Find the number of pairs of consecutive integers in the set $\{1026, 1027, 1028, \ldots, 2026\}$ such that when these two integers are added, no carrying occurs in any digit. 2. **Understanding the problem:** Two consecutive integers are $n$ and $n+1$. We want to add $n + (n+1) = 2n + 1$ without any digit carrying during the addition of $n$ and $n+1$. 3. **Key insight:** No carrying means that for each digit position, the sum of the digits of $n$ and $n+1$ in that position is less than 10. 4. **Analyzing the digits:** Since $n$ and $n+1$ differ by 1, the only digit that changes is the units digit. For no carrying: - The units digit of $n$ plus the units digit of $n+1$ must be less than 10. 5. **Units digit condition:** Let the units digit of $n$ be $d$. Then the units digit of $n+1$ is: - If $d < 9$, then units digit of $n+1$ is $d+1$. - If $d = 9$, then units digit of $n+1$ is 0, but this causes carrying. 6. **Sum of units digits:** For no carry in units digit: $$d + (d+1) < 10 \implies 2d + 1 < 10 \implies 2d < 9 \implies d < 4.5$$ So $d$ can be $0,1,2,3,4$. 7. **Tens digit and higher:** Since $n$ and $n+1$ differ only in the units digit, all other digits are the same, so their sum doubles the digit without carry. 8. **Check tens digit:** For no carry in tens digit, the sum of the tens digits of $n$ and $n+1$ must be less than 10. Since digits are the same, sum is $2 \times$ (tens digit). So tens digit must be less than 5. 9. **Similarly for hundreds and thousands digits:** Each digit $x$ must satisfy $2x < 10 \implies x < 5$. 10. **Range of $n$:** $1026$ to $2026$. 11. **Digits of $n$:** - Thousands digit: 1 or 2 - Hundreds digit: 0 or 1 or 2 - Tens digit: 2 or 3 or 4 or 5 or 6 or 7 or 8 or 9 - Units digit: 0 to 9 12. **From step 9, digits must be less than 5 to avoid carry:** - Thousands digit: must be 1 (since 2 is not less than 5, but 2 < 5 is true, so 2 is allowed) - Hundreds digit: must be less than 5 (0,1,2,3,4 allowed) - Tens digit: must be less than 5 (0,1,2,3,4 allowed) - Units digit: $d$ in $\{0,1,2,3,4\}$ from step 6 13. **Check thousands digit:** 1 or 2 both less than 5, so allowed. 14. **Therefore, the digits of $n$ must satisfy:** - Thousands digit: 1 or 2 - Hundreds digit: 0 to 4 - Tens digit: 0 to 4 - Units digit: 0 to 4 15. **Count numbers $n$ in $[1026,2026]$ with these digit restrictions and units digit $d \in \{0,1,2,3,4\}$:** 16. **Break into two parts:** - From 1026 to 1999 (thousands digit = 1) - From 2000 to 2026 (thousands digit = 2) 17. **For thousands digit = 1:** - Hundreds digit: 0 to 4 - Tens digit: 0 to 4 - Units digit: 0 to 4 18. **Range 1026 to 1999:** - Minimum is 1026, so hundreds digit at least 0, tens digit at least 2 (since 1026 has tens digit 2), units digit at least 6 for 1026. 19. **But units digit must be in $\{0,1,2,3,4\}$, so 6 is not allowed. So start from 1026, but units digit restriction excludes 6,7,8,9. So first valid number with units digit $\leq 4$ and tens digit $\geq 2$ is 1020 (units digit 0), but 1020 < 1026, so invalid. 20. **So for thousands=1, hundreds=0, tens=2, units=0 to 4, numbers less than 1026 are invalid. So valid numbers start from 1030 (units digit 0), 1031, 1032, 1033, 1034, etc. But 1030 > 1026, so valid. 21. **Count all numbers with thousands=1, hundreds=0 to 4, tens=0 to 4, units=0 to 4, and number $\geq 1026$:** 22. **Calculate total numbers with these digits:** - Hundreds digit: 0 to 4 (5 options) - Tens digit: 0 to 4 (5 options) - Units digit: 0 to 4 (5 options) - Total: $5 \times 5 \times 5 = 125$ 23. **Exclude numbers less than 1026:** - Numbers with hundreds=0, tens=0 or 1, units=0 to 4 are less than 1026. - Count these: - Hundreds=0 (1 option) - Tens=0 or 1 (2 options) - Units=0 to 4 (5 options) - Total: $1 \times 2 \times 5 = 10$ 24. **So valid numbers with thousands=1 are $125 - 10 = 115$ numbers.** 25. **For thousands digit = 2:** - Numbers from 2000 to 2026 - Hundreds digit: 0 to 4 - Tens digit: 0 to 4 - Units digit: 0 to 4 26. **Check which numbers in 2000 to 2026 satisfy these digit restrictions:** - 2000 to 2026 means hundreds digit 0 or 1 or 2 - Tens digit 0 or 1 or 2 - Units digit 0 to 6 27. **But units digit must be 0 to 4, so units digit 5 or 6 excluded. Tens digit must be 0 to 4, so tens digit 0 to 2 allowed. 28. **List all numbers from 2000 to 2026 with units digit 0 to 4:** - 2000, 2001, 2002, 2003, 2004 - 2010, 2011, 2012, 2013, 2014 - 2020, 2021, 2022, 2023, 2024 29. **Count these:** 5 numbers per tens digit (0,1,2) = $3 \times 5 = 15$ numbers. 30. **Check hundreds digit for these numbers:** - 2000 to 2004: hundreds digit 0 - 2010 to 2014: hundreds digit 1 - 2020 to 2024: hundreds digit 2 All less than 5, so allowed. 31. **Total valid numbers with thousands=2 is 15.** 32. **Total valid numbers $n$ in $[1026,2026]$ with digit restrictions is $115 + 15 = 130$.** 33. **Each such $n$ forms a pair $(n, n+1)$ with no carrying when added.** **Final answer:** $$\boxed{130}$$