The sum of all two digit positive numbers which when divided by 7 yield 2 or 5 as remainder is
16 + 23 + 30 + ...........+ 93 = 6 [16 + 93] = 6 × 109 = 654
and 12 + 19 + 26 + ........... + 96 = [96 + 12] = 13 × 54 = 702
_________
+ 1356
We are to find the sum of all two-digit positive numbers that leave a remainder of 2 or 5 when divided by 7.
Step 1: Identify the numbers
A number leaves remainder 2 or 5 when divided by 7. This means the number is congruent to 2 or 5 modulo 7. So, the numbers are of the form 7k + 2 or 7k + 5, where k is an integer.
Step 2: Find the range for two-digit numbers
The smallest two-digit number is 10 and the largest is 99.
For numbers of the form 7k + 2:
10 ≤ 7k + 2 ≤ 99 ⇒ 8 ≤ 7k ≤ 97 ⇒ k ≥ 2 (since 7*2=14 ≥8) and k ≤ 13 (since 7*13=91 ≤97). So k from 2 to 13.
Similarly, for 7k + 5:
10 ≤ 7k + 5 ≤ 99 ⇒ 5 ≤ 7k ≤ 94 ⇒ k ≥ 1 (since 7*1=7 ≥5) and k ≤ 13 (since 7*13=91 ≤94). So k from 1 to 13.
Step 3: List the numbers
For 7k+2: k=2 to 13 gives numbers: 16, 23, 30, 37, 44, 51, 58, 65, 72, 79, 86, 93.
For 7k+5: k=1 to 13 gives numbers: 12, 19, 26, 33, 40, 47, 54, 61, 68, 75, 82, 89, 96.
Note: There is no overlap since a number cannot be both 2 and 5 mod 7.
Step 4: Count the numbers
From 7k+2: 12 numbers (k=2 to 13 inclusive).
From 7k+5: 13 numbers (k=1 to 13 inclusive).
Total numbers: 12 + 13 = 25.
Step 5: Compute the sum
We can compute the sum of each arithmetic sequence and add them.
Sequence 1: 7k+2 for k=2 to 13. This is an arithmetic progression (AP) with first term a₁ = 7*2+2=16, last term l₁ = 7*13+2=93, number of terms n₁=12.
Sum S₁ = n₁/2 * (a₁ + l₁) = 12/2 * (16 + 93) = 6 * 109 = 654.
Sequence 2: 7k+5 for k=1 to 13. AP with first term a₂ = 7*1+5=12, last term l₂ = 7*13+5=96, number of terms n₂=13.
Sum S₂ = n₂/2 * (a₂ + l₂) = 13/2 * (12 + 96) = 13/2 * 108 = 13 * 54 = 702.
Total sum S = S₁ + S₂ = 654 + 702 = 1356.
Final Answer: 1356
Arithmetic Progression (AP): A sequence where the difference between consecutive terms is constant. The sum of the first n terms is given by S = n/2 * (first term + last term) or S = n/2 * [2a + (n-1)d], where a is the first term and d is the common difference.
Modular Arithmetic: Numbers that leave the same remainder when divided by a number are said to be congruent modulo that number. Here, we used numbers ≡ 2 mod 7 and ≡ 5 mod 7.