1. **Problem statement:** Find the number of pairs of consecutive integers in the set $\{2026, 2027, 2028, \ldots, 3026\}$ such that no carrying is required when adding the two integers. 2. **Understanding the problem:** When adding two consecutive integers $n$ and $n+1$, no carrying means that for each digit place, the sum of the digits of $n$ and $n+1$ is less than 10. 3. **Key insight:** Since the numbers are consecutive, the digits differ only in the last digit (units place) or may affect higher digits if the last digit is 9. 4. **Condition for no carrying:** For each digit place, the sum of the digit of $n$ and the digit of $n+1$ must be less than 10. 5. **Focus on the units digit:** Let the units digit of $n$ be $d$. Then the units digit of $n+1$ is $(d+1) \bmod 10$. - If $d \neq 9$, then units digits are $d$ and $d+1$, sum is $2d+1$ which must be less than 10. - If $d=9$, then units digits are 9 and 0, sum is 9 which is less than 10. 6. **Check the sum for units digit:** For $d=0$ to $8$, sum is $d + (d+1) = 2d+1 < 10$. - For $d=4$, sum is $2\times4+1=9<10$ (valid) - For $d=5$, sum is $11$ (not valid) Actually, let's check all $d$: $d=0$: sum=1 <10 valid $d=1$: sum=3 <10 valid $d=2$: sum=5 <10 valid $d=3$: sum=7 <10 valid $d=4$: sum=9 <10 valid $d=5$: sum=11 not valid $d=6$: sum=13 not valid $d=7$: sum=15 not valid $d=8$: sum=17 not valid $d=9$: sum=9 <10 valid So units digit $d$ must be in $\{0,1,2,3,4,9\}$. 7. **Check tens digit:** When units digit is 9, adding 1 causes a carry to tens digit, so tens digit of $n+1$ is tens digit of $n$ plus 1. - For no carrying, sum of tens digits must be less than 10. - If units digit is 9, tens digit of $n+1$ is tens digit of $n$ plus 1, so sum of tens digits is $t + (t+1) = 2t+1 < 10$. - For units digit not 9, tens digits are equal, sum is $2t < 10$. 8. **Generalizing for all digits:** For digits other than units digit: - If units digit is not 9, digits are equal, sum is $2d_i < 10$ so $d_i \leq 4$. - If units digit is 9, next digit increases by 1, sum is $2d_i + 1 < 10$ so $d_i \leq 4$. 9. **Apply to the range 2026 to 3026:** - Numbers are four-digit: thousands, hundreds, tens, units. - Thousands digit is 2 or 3. - For no carrying, digits except units must be $\leq 4$. - Thousands digit 3 is allowed only if $3 \leq 4$ (true). 10. **Check thousands digit:** - For units digit not 9, sum of thousands digits is $2 \times 2 = 4 < 10$ or $2 \times 3 = 6 < 10$ valid. - For units digit 9, sum is $2 \times 2 + 1 = 5 < 10$ or $2 \times 3 + 1 = 7 < 10$ valid. 11. **Check hundreds and tens digits:** Must be $\leq 4$. 12. **Count valid pairs:** - Total numbers: $3026 - 2026 + 1 = 1001$. - Pairs: $1000$ pairs of consecutive integers. - For each pair, check units digit of first number in pair. - Units digit must be in $\{0,1,2,3,4,9\}$. 13. **Count numbers in range with units digit in $\{0,1,2,3,4,9\}$:** - Units digits cycle every 10 numbers. - Number of full cycles: $\lfloor \frac{1001}{10} \rfloor = 100$. - Each cycle has 6 valid units digits. - So from full cycles: $100 \times 6 = 600$ valid numbers. - Remaining numbers: $1001 - 1000 = 1$ number (3026), units digit 6 (not valid). - So total valid numbers with valid units digit: 600. 14. **Check if all these 600 numbers satisfy other digit conditions:** - Since range is from 2026 to 3026, hundreds and tens digits vary. - We must ensure hundreds and tens digits $\leq 4$. - Let's analyze the hundreds and tens digits for numbers in range: - Numbers from 2026 to 2999 have thousands digit 2. - Numbers from 3000 to 3026 have thousands digit 3. - For numbers 2026 to 2999, hundreds digit ranges from 0 to 9. - For no carrying, hundreds digit $\leq 4$. - So numbers with hundreds digit $>4$ are invalid. - Similarly for tens digit. 15. **Count numbers with hundreds digit $\leq 4$ and tens digit $\leq 4$:** - Hundreds digit ranges from 0 to 9. - Tens digit ranges from 0 to 9. - For no carrying, both $\leq 4$. - So hundreds digit in $\{0,1,2,3,4\}$ and tens digit in $\{0,1,2,3,4\}$. 16. **Count numbers in range 2026 to 2999 with these conditions:** - Thousands digit fixed at 2. - Hundreds digit: 0 to 4 (5 options). - Tens digit: 0 to 4 (5 options). - Units digit: in $\{0,1,2,3,4,9\}$ (6 options). - Total numbers: $5 \times 5 \times 6 = 150$. - But must be within 2026 to 2999. - Minimum number with hundreds=0, tens=0, units=0 is 2000 (less than 2026), so exclude numbers less than 2026. - Minimum valid number is 2026. - So count numbers from 2026 to 2999 with above digit restrictions. 17. **Count numbers from 2026 to 2999 with hundreds digit 0 to 4, tens digit 0 to 4, units digit in $\{0,1,2,3,4,9\}$:** - For hundreds digit 0: numbers 2000 to 2099. - Tens digit 0 to 4: 2000 to 2049. - Units digit in $\{0,1,2,3,4,9\}$. - Numbers less than 2026 excluded. - So for hundreds=0, tens=2, units $\geq 6$ excluded. - Count numbers from 2026 to 2049 with units digit in $\{0,1,2,3,4,9\}$. - Units digits in 2026 to 2049: - 2026 (units 6) no - 2027 (7) no - 2028 (8) no - 2029 (9) yes - 2030 (0) yes - 2031 (1) yes - 2032 (2) yes - 2033 (3) yes - 2034 (4) yes - 2035 (5) no - 2036 (6) no - 2037 (7) no - 2038 (8) no - 2039 (9) yes - 2040 (0) yes - 2041 (1) yes - 2042 (2) yes - 2043 (3) yes - 2044 (4) yes - 2045 (5) no - 2046 (6) no - 2047 (7) no - 2048 (8) no - 2049 (9) yes Count valid units digits: 2029,2030,2031,2032,2033,2034,2039,2040,2041,2042,2043,2044,2049 = 13 numbers. 18. **For hundreds digit 1 to 4:** - Hundreds digit 1: 2100 to 2199 - Tens digit 0 to 4: 2100 to 2149 - Units digit in $\{0,1,2,3,4,9\}$: 6 options per tens digit - Tens digit 0 to 4: 5 options - Units digit 6 options - Total: $5 \times 6 = 30$ numbers per hundreds digit. - For hundreds digits 1,2,3,4: $4 \times 30 = 120$ numbers. 19. **Total valid numbers from 2026 to 2999:** - For hundreds=0: 13 numbers - For hundreds=1 to 4: 120 numbers - Total: $13 + 120 = 133$ numbers. 20. **For numbers 3000 to 3026:** - Thousands digit 3 (allowed) - Hundreds digit 0 (allowed) - Tens digit 0 to 2 (since max 3026) - Units digit in $\{0,1,2,3,4,9\}$ - Count numbers 3000 to 3026 with tens digit $\leq 2$ and units digit in $\{0,1,2,3,4,9\}$. - Tens digit 0: 3000 to 3009 - Units digit valid: 0,1,2,3,4,9 (6 numbers) - Tens digit 1: 3010 to 3019 - Units digit valid: 6 numbers - Tens digit 2: 3020 to 3026 - Units digit valid and within range: 3020(0),3021(1),3022(2),3023(3),3024(4),3025(5 no),3026(6 no) - Valid units digits: 0,1,2,3,4 (5 numbers) - Total: $6 + 6 + 5 = 17$ numbers. 21. **Total valid numbers in entire range:** - $133 + 17 = 150$ numbers. 22. **Each valid number corresponds to a pair $(n, n+1)$ with no carrying.** 23. **Final answer:** There are $\boxed{150}$ pairs of consecutive integers in the set $\{2026, \ldots, 3026\}$ such that no carrying is required when adding the two integers.